Study sheet: Real Analysis and Elementary Functions

📋 Course Outline

1. Logic, Quantifiers, and Proofs
2. Sets, Relations, and Functions
3. Integers, Combinations, and Real Numbers
4. Order, Absolute Value, and Bounds
5. Polynomials and Euclidean Division
6. Logarithms, Exponentials, and Powers
7. Trigonometric Functions and Equations
8. Algebraic Form of Complex Numbers
9. Conjugation and Complex Modulus
10. Trigonometric Form and Roots
11. Limits and Continuity
12. Continuity and Classical Limits
13. Asymptotes and Function Study
14. Differentiation and Its Applications
15. Vectors, Coordinates and Angles
16. Oriented Angles and Orthonormal Bases
17. Scalar Product and Determinant
18. Lines and Circles in the Plane

📖 1. Logic, Quantifiers, and Proofs

🔑 Key Concepts & Definitions

• Mathematical assertion : a statement that is unambiguously either true or false
• Quantifiers : The existential quantifier ∃x means that at least one x satisfies the assertion, the universal quantifier ∀x means that every x satisfies it, and ∃!x means that exactly one x satisfies it.

📝 Essential Points

• The conjunction A and B is true only when both assertions are true, whereas the inclusive disjunction A or B is true when at least one of A and B is true.

• The implication A ⇒ B requires B to be true when A is true, so A is sufficient for B and B is necessary for A.

• An implication A ⇒ B is equivalent to its contrapositive ¬B ⇒ ¬A.

• Negating a quantified assertion exchanges ∃ and ∀ and negates the final assertion, so ¬(∃x,A(x)) is equivalent to ∀x,¬A(x) and ¬(∀x,A(x)) is equivalent to ∃x,¬A(x).

• A direct proof assumes the hypothesis; a proof by contrapositive proves ¬B ⇒ ¬A; a proof by contradiction assumes ¬A and derives incompatible conclusions; a proof by cases proves the conclusion in every exhaustive case; and a counterexample disproves a universal assertion.

💡 Memory Hook

Assertion → quantifier → implication → proof

📖 2. Sets, Relations, and Functions

🔑 Key Concepts & Definitions

• Set inclusion : included in B, written A ⊂ B, when every element of A is also an element of B
• Function : a relation in which each element x of E is related to at most one element f(x) of F
• Function composition : For f:E→F and g:F→G, the composition g∘f is defined by $(g\circ f)(x)=g(f(x))$ on the set of x for which x belongs to Df and f(x) belongs to Dg.
• Cartesian product : the set of ordered pairs (a,b) such that a ∈ A and b ∈ B

★ Must-know

📌 The intersection A ∩ B contains elements belonging to both A and B, whereas the union A ∪ B contains elements belonging to at least one of A and B.

📌 An injective application gives distinct images to distinct inputs, a surjective application gives every target element at least one antecedent, and a bijective application is both injective and surjective.

Further detail

📌 For an application f:E→F, f is injective exactly when every equation f(x)=y has at most one solution in E, surjective exactly when it has at least one solution, and bijective exactly when it has exactly one solution.

💡 Memory Hook

A relation may associate several outputs, whereas a function associates at most one output to each input.

📖 3. Integers, Combinations, and Real Numbers

🔑 Key Concepts & Definitions

• Absolute value : The absolute value of a real number x is the nonnegative real number $|x|=\max(-x,x)$, equal to x when x≥0 and to −x when x<0.

📝 Essential Points

📌 For natural numbers a and b with b≠0, there is a unique pair of integers q and r such that a=bq+r and r<b; q and r are the quotient and remainder of Euclidean division.

• Every natural number n≥2 has a unique prime factorization n=p₁^α₁⋯pₖ^αₖ with p₁<⋯<pₖ prime and each exponent αᵢ a nonzero integer.

• A proof by induction establishes P(n) for every natural number by proving the initialization P(0) and the hereditary implication P(n)⇒P(n+1) for every n.

📐 Formula — The binomial formula is $(a+b)^n=\sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k$ for every natural number n.

📐 Formula — The number of combinations of p elements chosen from n is $\binom{n}{p}=\frac{n!}{p!(n-p)!}$ for p between 0 and n.

💡 Memory Hook

Division, factorization, and order properties provide the structure for arithmetic over number systems.

📖 4. Order, Absolute Value, and Bounds

🔑 Key Concepts & Definitions

• Order relation : The relation $x\le y\Longleftrightarrow y-x\in[0,+\infty[$ is a total order on $\mathbb R$ because it is reflexive, transitive, antisymmetric, and any two real numbers are comparable.
• Supremum : For a nonempty majorized subset $A$ of $\mathbb R$, $S=\sup A$ means that $S$ is a majorant of $A$ and every majorant of $A$ is at least $S$.
• Integer part : the unique integer satisfying $\lfloor x\rfloor\le x<\lfloor x\rfloor+1$

📝 Essential Points

• Adding or subtracting the same real number preserves an inequality, multiplication by a nonnegative number preserves its direction, and multiplication by a nonpositive number reverses its direction.

• For $a\ge0$, the inequalities $|x|\le a$ and $|x|>a$ are respectively equivalent to $-a\le x\le a$ and $x<-a\text{ or }x>a$.

• A maximum or minimum must belong to the set, whereas a supremum or infimum may be approached without belonging to the set.

💡 Memory Hook

Maximum belongs to the set; supremum need not.

📖 5. Polynomials and Euclidean Division

🔑 Key Concepts & Definitions

• Polynomial : A real polynomial can be represented by an infinite real coefficient sequence that is zero from some rank onward, with addition and multiplication defined coefficient by coefficient and by convolution.
• Euclidean division : For polynomials $A$ and nonzero $B$, there is a unique pair $Q,R$ such that $A=BQ+R$ and $\deg(R)<\deg(B)$.

📝 Essential Points

📐 Formula — For nonzero polynomials, the degree satisfies $\deg(A+B)\le\max(\deg A,\deg B)$ and $\deg(AB)=\deg A+\deg B$.

• Polynomial Euclidean division repeatedly cancels the dominant term of the current dividend by subtracting a suitable multiple of the divisor until the remainder has smaller degree than the divisor.

📌 A polynomial $B$ divides a polynomial $A$ if and only if the remainder of the Euclidean division of $A$ by $B$ is zero.

📌 A real number $a$ is a root of a polynomial $P$ if and only if $X-a$ divides $P$.

• A nonzero real polynomial of degree $n\ge0$ has at most $n$ distinct real roots.

💡 Memory Hook

Divide, identify the remainder, then test divisibility.

📖 6. Logarithms, Exponentials, and Powers

🔑 Key Concepts & Definitions

• Natural logarithm : The natural logarithm is defined on $]0,+\infty[$ by $\ln(x)=\int_1^x\frac{1}{t}\,dt$, with $\ln(1)=0$ and $\ln'(x)=\frac1x$.
• Exponential function : The exponential function is the inverse of the natural logarithm, satisfies $\exp(0)=1$, $\exp(1)=e$, and has derivative $\exp'(x)=\exp(x)>0$.

📝 Essential Points

📐 Formula — For positive $a,b$, logarithms satisfy $\ln(ab)=\ln a+\ln b$, $\ln(a/b)=\ln a-\ln b$, and $\ln(a^n)=n\ln a$.

• The natural logarithm is a strictly increasing bijection from $]0,+\infty[$ onto $\mathbb R$, and there is a unique positive number $e$ such that $\ln e=1$.

📐 Formula — For $a>0$ and $\alpha\in\mathbb R$, the real power is defined by $a^\alpha=\exp(\alpha\ln a)$.

📐 Formula — For real $\alpha$ and positive $x$, the power function satisfies $\frac{d}{dx}x^\alpha=\alpha x^{\alpha-1}$.

💡 Memory Hook

The logarithm turns products into sums; the exponential reverses that process.

📖 7. Trigonometric Functions and Equations

🔑 Key Concepts & Definitions

• Sine and cosine : On the unit circle, $\cos x$ and $\sin x$ are respectively the abscissa and ordinate of the point associated with the oriented angle $x$, and they satisfy $\cos^2x+\sin^2x=1$.
• Tangent : The tangent function is defined by $\tan x=\frac{\sin x}{\cos x}$ on $\mathbb R\setminus\{\frac\pi2+k\pi:k\in\mathbb Z\}$.

★ Must-know

• The sine and cosine functions are continuous, differentiable, and periodic with period $2\pi$, with $\sin'(x)=\cos x$ and $\cos'(x)=-\sin x$.

📐 Formula — On its domain, the tangent function has period $\pi$ and derivative $\tan'(x)=1+\tan^2x=\frac1{\cos^2x}$.

📐 Formula — The addition formulas are $\cos(a+b)=\cos a\cos b-\sin a\sin b$ and $\sin(a+b)=\sin a\cos b+\cos a\sin b$.

Further detail

• To solve a trigonometric equation or inequality, one rewrites it using identities or a substitution, solves the resulting algebraic or basic trigonometric condition, and restricts the solution set to the requested interval.

💡 Memory Hook

The unit circle turns angles into coordinates: cosine horizontally and sine vertically.

📖 8. Algebraic Form of Complex Numbers

🔑 Key Concepts & Definitions

• Complex number : has the algebraic form z = x + iy with x,y ∈ R, where x = Re(z) is its real part and y = Im(z) is its imaginary part.
• Complex plane : represents z = x + iy by the point M(x,y), or equivalently by the vector x⃗i + y⃗j, in an orthonormal coordinate system.

★ Must-know

📐 Formula — For z = x + iy and z' = x' + iy', addition and multiplication are given by $z+z'=(x+x')+i(y+y')$ and $zz'=(xx'-yy')+i(xy'+x'y).$

Further detail

📐 Formula — If A and B have affixes zA and zB, then the affix of the vector AB is $\operatorname{aff}(\overrightarrow{AB})=z_B-z_A.$

💡 Memory Hook

Real part lies on the real axis; imaginary part lies on the imaginary axis.

📖 9. Conjugation and Complex Modulus

🔑 Key Concepts & Definitions

• Complex conjugate : the complex number z̄ = x − iy.
• Complex modulus : the nonnegative real number $|z|=\sqrt{x^2+y^2}.$

★ Must-know

📐 Formula — A complex number and its conjugate satisfy $z\bar z=|z|^2=x^2+y^2.$

Further detail

📐 Formula — For z ≠ 0, its inverse is $\frac1z=\frac{\bar z}{|z|^2}=\frac{x-iy}{x^2+y^2}.$

💡 Memory Hook

A complex number and its conjugate mirror each other across the real axis.

📖 10. Trigonometric Form and Roots

🔑 Key Concepts & Definitions

• Argument : any real θ such that $z=|z|(\cos\theta+i\sin\theta),$ and all its arguments are θ + 2kπ with k ∈ Z.

📝 Essential Points

📐 Formula — Every nonzero complex number has the trigonometric form $z=re^{i\theta},$ where r = |z| and θ is an argument of z modulo 2π.

📐 Formula — For nonzero complex numbers z and z', arguments satisfy $\arg(zz')=\arg(z)+\arg(z')\ [2\pi]$ and $\arg\left(\frac z{z'}\right)=\arg(z)-\arg(z')\ [2\pi].$

📐 Formula — If z = reiθ with r > 0, then its n-th roots are $Z_k=\sqrt[n]{r}e^{i(\theta+2k\pi)/n},\quad k=0,\ldots,n-1,$ so z has exactly n distinct n-th roots.

💡 Memory Hook

Modulus gives the radius, argument gives the angle, and roots divide the angle.

📖 11. Limits and Continuity

🔑 Key Concepts & Definitions

• Finite limit : has finite limit ℓ at a when every ε > 0 admits α > 0 such that x ∈ D and |x − a| ≤ α imply |f(x) − ℓ| ≤ ε.
• Continuity : is defined at a and $\lim_{x\to a}f(x)=f(a).$

★ Must-know

📌 A two-sided limit at a exists when the left-hand and right-hand limits exist and are equal, with the common value also equal to f(a) when f is defined at a.

📌 If f is continuous on an interval and takes values of opposite signs, then there exists x in the interval such that f(x) = 0.

Further detail

📌 If f is continuous on a segment [a,b], then f([a,b]) is a segment [m,M], so f is bounded and attains both its minimum m and maximum M.

💡 Memory Hook

Matching one-sided limits produces a two-sided limit; matching the function value produces continuity.

📖 12. Continuity and Classical Limits

★ Must-know

• If h(x) ≤ f(x) ≤ g(x) near +∞ and both h and g tend to ℓ, then f tends to ℓ by the squeeze theorem; the same result applies near −∞ or near a finite point.

• Sums, products, scalar multiples and reciprocals of functions continuous at a point are continuous there, provided the reciprocal function's denominator does not vanish near that point.

• If f is continuous at a, f(I) is contained in J, and g is continuous at f(a), then the composition g∘f is continuous at a.

• For a reduced rational function P(x)/Q(x) with numerator degree n and denominator degree p, the limit at ±∞ is 0 if n<p, infinite in the leading-term direction if n>p, and a_n/b_p if n=p.

Further detail

• For a polynomial of degree n with leading coefficient a_n, the limit at ±∞ is determined by the leading term a_nx^n.

💡 Memory Hook

Bounds with the same limit force the middle function to share it.

📖 13. Asymptotes and Function Study

🔑 Key Concepts & Definitions

• Oblique asymptote : a line y=ax+b is an asymptote at +∞ to the graph of f if f(x)=ax+b+φ(x) with φ(x) tending to 0 as x tends to +∞

★ Must-know

📐 Formula — If an oblique asymptote exists at +∞, its coefficients satisfy $a=\lim_{x\to+\infty}\frac{f(x)}{x}$ and $b=\lim_{x\to+\infty}(f(x)-ax)$.

• For f(x)=(x+2)^2/[3(x+1)], the vertical asymptote is x=−1 and the oblique asymptote at both +∞ and −∞ is y=x/3+1.

Further detail

• For f(x)=2x+3+e^{-x}, the line y=2x+3 is an asymptote at +∞ because e^{-x} tends to 0.

💡 Memory Hook

Vertical asymptotes come from infinite limits at a point; horizontal or oblique asymptotes come from behavior at infinity.

📖 14. Differentiation and Its Applications

🔑 Key Concepts & Definitions

• Derivative : A function f is differentiable at a if the limit $f'(a)=\lim_{h\to0}\frac{f(a+h)-f(a)}{h}$ exists and is finite.

📝 Essential Points

📐 Formula — The tangent to the graph of f at (a,f(a)) has equation $y=f(a)+f'(a)(x-a)$.

📌 If f is differentiable at a, then f is continuous at a, but continuity at a does not imply differentiability there, as shown by |x| at 0.

📐 Formula — For differentiable functions u and v, the product and quotient rules are $(uv)'=u'v+uv'$ and $\left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^2}$ when v is nonzero.

• If f has a local extremum at an interior point c and is differentiable there, then f'(c)=0.

• Rolle's theorem states that a function continuous on [a,b], differentiable on ]a,b[, and satisfying f(a)=f(b) has at least one c in ]a,b[ such that f'(c)=0.

• The mean value theorem states that a function continuous on [a,b] and differentiable on ]a,b[ has some c in ]a,b[ satisfying $f(b)-f(a)=(b-a)f'(c)$.

💡 Memory Hook

Rate of change → derivative → tangent → extrema and variation.

📖 15. Vectors, Coordinates and Angles

🔑 Key Concepts & Definitions

• Vector norm : the nonnegative length AB, and a vector of norm 1 is called unitary
• Collinear vectors : Two nonzero vectors are collinear when they have the same direction; equivalently, one is a real scalar multiple of the other.

★ Must-know

📌 In a Cartesian frame (O,i,j) with noncollinear basis vectors, every point and every vector has a unique pair of real coordinates.

📌 The geometric angle between two nonzero vectors is the shortest angular distance in [0,π], whereas an oriented angle has measures defined modulo 2π after choosing an orientation of the plane.

Further detail

📐 Formula — For vectors with coordinates (x,y) and (x',y'), vector addition and scalar multiplication are given by $(x,y)+(x',y')=(x+x',y+y')$ and $\lambda(x,y)=(\lambda x,\lambda y)$.

💡 Memory Hook

Geometric angles lie in [0,π], whereas oriented angles are defined modulo 2π.

📖 16. Oriented Angles and Orthonormal Bases

🔑 Key Concepts & Definitions

• Principal measure of an oriented angle : the signed angular gap α, positive in the direct direction and negative otherwise

★ Must-know

📌 All measures of an oriented angle with principal measure α are of the form $\alpha+2k\pi$ with $k\in\mathbb{Z}$, and reversing the orientation changes α to −α modulo $2\pi$.

• For three nonzero vectors, oriented angles satisfy the Chasles relation $({\vec u},{\vec v})=({\vec u},{\vec w})+({\vec w},{\vec v})\ [2\pi]$.

📌 Two nonzero vectors are orthogonal if and only if their oriented angle is $\frac{\pi}{2}\ [\pi]$, whereas they are collinear if and only if their oriented angle is $0\ [\pi]$.

Further detail

📐 Formula — In a direct orthonormal basis, a nonzero vector with coordinates $(x,y)$ and angle α from the first basis vector satisfies $x=\|\vec u\|\cos\alpha$ and $y=\|\vec u\|\sin\alpha$.

💡 Memory Hook

Direct orientation gives +π/2; reversing orientation gives −π/2.

📖 17. Scalar Product and Determinant

🔑 Key Concepts & Definitions

• Scalar product : $\vec u\cdot\vec v=\|\vec u\|\,\|\vec v\|\cos(\vec u,\vec v)$, and it is zero if either vector is zero
• Determinant : $\det(\vec u,\vec v)=\|\vec u\|\,\|\vec v\|\sin(\vec u,\vec v)$, and it is zero if either vector is zero

★ Must-know

📌 Two vectors are orthogonal if and only if their scalar product is zero, whereas they are collinear if and only if their determinant is zero.

📐 Formula — For vectors with coordinates $(x,y)$ and $(x',y')$ in an orthonormal basis, $\vec u\cdot\vec v=xx'+yy'$ and, in a direct basis, $\det(\vec u,\vec v)=xy'-yx'$.

📐 Formula — The orthogonal projection of a vector \vec u onto the line generated by a nonzero vector \vec v is $p(\vec u)=\frac{\vec u\cdot\vec v}{\|\vec v\|^2}\vec v$.

Further detail

📐 Formula — The area of triangle ABC is $\mathcal A(ABC)=\frac12\left|\det(\overrightarrow{AB},\overrightarrow{AC})\right|$.

💡 Memory Hook

The scalar product measures alignment; the determinant measures oriented area.

📖 18. Lines and Circles in the Plane

🔑 Key Concepts & Definitions

• Affine plane : the set of points M such that $\overrightarrow{AM}=\alpha\vec u+\beta\vec v$ for some $(\alpha,\beta)\in\mathbb R^2
• Point-to-plane distance : the length of the orthogonal projection segment AH, where H is the projection of A onto P

📝 Essential Points

• A line through A with direction vector \vec u has the parametric representation $x=x_A+\alpha u_x,\ y=y_A+\alpha u_y$ with $\alpha\in\mathbb R$.

📐 Formula — A line with Cartesian equation $ax+by+c=0$ has normal vector $(a,b)$ and direction vector $(-b,a)$ when $(a,b)\ne(0{,}0)$.

📐 Formula — The distance from A(x_A,y_A) to the line $ax+by+c=0$ is $d(A,D)=\frac{|ax_A+by_A+c|}{\sqrt{a^2+b^2}}$.

📐 Formula — In an orthonormal coordinate system, the circle with centre A(x_A,y_A) and radius R has equation $(x-x_A)^2+(y-y_A)^2=R^2$.

📐 Formula — In an orthonormal spatial basis, a vector with coordinates $(x,y,z)$ has norm $\|\vec u\|=\sqrt{x^2+y^2+z^2}$ and the scalar product of vectors with coordinates $(x,y,z)$ and $(x',y',z')$ is $xx'+yy'+zz'$.

• A spatial line uses one real parameter in its three coordinate equations, whereas a spatial plane uses two real parameters in its three coordinate equations.

📌 A spatial plane has a Cartesian equation $ax+by+cz+d=0$ with nonzero normal vector $(a,b,c)$, and two such equations define the same plane exactly when their coefficient quadruples are proportional.

📌 Two spatial planes intersect in a line when their normal vectors are not proportional, coincide when their full coefficient quadruples are proportional, and are strictly parallel when their normal vectors are proportional but their full quadruples are not.

• The intersection of a plane and a line is found by solving the system formed by the plane equation and the two Cartesian equations of the line, using Gaussian elimination.

• The intersection of a plane and a line is either one point when they are secant, the entire line when the line is included in the plane, or the empty set when the line and plane are strictly parallel.

• If two lines in space intersect, they are coplanar.

• Two coplanar lines with parallel direction vectors are parallel, whereas two coplanar lines with non-parallel direction vectors are secant and have exactly one intersection point.

📐 Formula — For a plane $P: ax+by+cz+d=0$ and a point $A(x_A,y_A,z_A)$, the distance is $d(A,P)=\frac{|ax_A+by_A+cz_A+d|}{\sqrt{a^2+b^2+c^2}}$.

📐 Formula — For a point A and a line D(B, u), the distance is $d(A,D)=\sqrt{\|\overrightarrow{AB}\|^2-\frac{(\vec u\cdot\overrightarrow{AB})^2}{\|\vec u\|^2}}$.

💡 Memory Hook

Parametric form → Cartesian equation → intersections and distances.

📊 Synthesis Tables

Key Elementary Functions

Function	Domain	Derivative	Period or range
ln x	]0,+∞[	$1/x$	Bijection onto $\mathbb R$
exp x	$\mathbb R$	$\exp x$	Bijection onto $]0,+∞[$
sin x	$\mathbb R$	$\cos x$	Period $2\pi$
cos x	$\mathbb R$	$-\sin x$	Period $2\pi$
tan x	$\mathbb R\setminus\{\pi/2+k\pi\}$	$1/\cos^2x$	Period $\pi$

Key differentiation results

Result	Hypotheses	Conclusion
Rolle's theorem	Continuous on [a,b], differentiable on ]a,b[, f(a)=f(b)	There exists c with f'(c)=0
Mean value theorem	Continuous on [a,b], differentiable on ]a,b[	There exists c with f(b)-f(a)=(b-a)f'(c)
Extremum condition	Differentiable at an interior local extremum	f'(c)=0