5. Functions

Def.: relation with exactly one output per input (well-definedness).

$$\begin{array}{rl}f:domain/\mathbb{D}(f)& \to range/\mathbb{W}(f)\\ x& \mapsto y=f(x)\end{array}$$

Input: independent variable Output: dependent variable

Restriction $f{|}_{D^{\prime }}:D^{\prime }\to \mathbb{R}\text{with }f{|}_{D^{\prime }}(x)=f(x)\forall x\in D^{\prime }$. Can have different properties when analyzed $⟹$ different function.

Bounded: $|f(x)|\le M\forall x\in \mathbb{D}(f)$. Bounded above and bounded below are analogous.

Uneven: $\forall x\in \mathbb{R}:f(-x)=-f(x)$ Even: $\forall x\in \mathbb{R}:f(-x)=f(x)$

Convex: curved to the left Concave: curved to the right

Theorem

Every strictly monotone function is injective

Limits

$X=\mathbb{D}(f)\cap (x_0-\delta ,x_0+\delta )$, i.e. all elements in the interval around $x_0$ that are in $\mathbb{D}(f)$. If $X\ne \varnothing$ for all $\delta >0$ (func defined in vicinity of $x$), then $L$ is a limit iff:

$$\forall \epsilon >0\exists \delta >0\text{ s.t. }\forall x\in X:|f(x)-L|<\epsilon$$

One-Sided Limits

Right-sided limit:

• $X=\mathbb{D}(f)\cap [x_0,x_0+\delta )$, same condition for $L$ as before
• $\underset{\begin{array}{c}x\ge x_0\\ x\to x_0\end{array}}{\lim }f(x)=L$

Left-sided limit: $X=\mathbb{D}(f)\cap (x_0-\delta ,x_0]$

In Infinity

In positive infinity: $X=\mathbb{D}(f)\cap (\delta ,\infty )$

In negative infinity: $X=\mathbb{D}(f)\cap (-\infty ,-\delta )$

Improper Limits

Can be either in infinity or at a specific $x_0$.

To positive infinity: $\forall N>0\exists \delta >0\text{ s.t. }\forall x\in X:f(x)\ge N$

To negative infinity: $\forall N>0\exists \delta >0\text{ s.t. }\forall x\in X:f(x)\le N$

Continuity

$$\forall \epsilon >0\exists \delta >0\text{ s.t. }\forall x\in I:|x-x_0|<\delta \to |f(x)-f(x_0)|<\epsilon$$

Intuitively:

• Choose any point $x_0$
• Create a rectangle of width $\delta$ to the left and right. Chose the height s.t. the rectangle contains the function values in the interval.
• If you make $\delta$ infinitely small, the height must be infinitely small as well.

Right-continuous: $\underset{\begin{array}{c}x\to x_0\\ x\ge x_0\end{array}}{\lim }f(x)=f(x_0)$ Left-continuous: completely analogous

Theorem

A func $f$ is continuous at $x_0$ $⟺$ for every sequence $(x_n{)}_{n\in {\mathbb{N}}_0}$ with $x_n\to x_0$ the sequence $(f(x_n{)}_{n\in {\mathbb{N}}_0})$ converges toward $f(x_0)$.

**Calculation rules:

• If $f$ continuous, $c$ const. $⟹$ $c\cdot f$ continuous
• If $f,g$ are continuous $⟹$ $f+g,f-g,f\cdot g,f\circ g$ continuous. $\underset{x\to x_0}{\lim }g(f(x))=g\left(\underset{x\to x_0}{\lim }f(x)\right)$.
• If $g\ne 0$: $\frac{f}{g}$ continuous

Theorem

$I=$ interval and $f:I\to J$ continuous and bijective $⟹$ J is interval and $f^{-1}:J\to I$ is continuous.

Theorem

Let $f:[a,b]\to \mathbb{R}$ continuous. Let $c$ be between $f(a)$ and $f(b)$. $⟹x\in [a,b]$ with $f(x)=c$

If $f(a)<0$ and $f(b)>b$ or v.v: $f$ has at least one x-intercept between $a$ and $b$.

Compact interval: bounded (finite length) and closed (endpoints included)

Lemma

$[a,b]=$ compact interval. $(x_n{)}_{n\in {\mathbb{N}}_0}$ in $[a,b]$. $⟹$ exists subsequence $(x_{n_k}{)}_{k\in {\mathbb{N}}_0}$ with:

$$\underset{k\to \infty }{\lim }{x}_{n_k}=x_0\text{with}{x}_0\in [a,b]$$

Extrema: $f:D\subset \mathbb{R}\to \mathbb{R}$

• $x_0\in D$ and $f(x_0)\ge f(x)\forall x\in D$ $⟹$ local maximum
• $x_0\in D$ and $f(x_0)\le f(x)\forall x\in D$ $⟹$ local minimum

Theorem

$f:I\subset \mathbb{R}\to \mathbb{R}=$ continuous and $I$ compact $⟹$ $f$ bounded and there exists a maximum and minimum in $I$ $⟺$ exists $x_{min}\in I$ and $x_{max}\in I$ with $\forall x\in I:f(x_{min})\le f(x)\le f(x_{max})$