Continuity of Real Functions

Continuity

Definition: Continuity of Real Functions

Let $f:\mathcal{D}\subseteq \mathbb{R}\to \mathbb{R}$ be a real function and let $c\in \mathbb{R}$.

We say that $f$ is continuous at $c$ if $f$‘s limit at $c$ is equal to $f(c)$:

$$\underset{x\to c}{\lim }f(x)=f(c)$$

If $f$ is continuous at every $c\in S\subseteq \mathcal{D}$, then we say that $f$ is continuous on $S$. If $S=\mathcal{D}$, then we simply say that $f$ is continuous.

Theorem: Operations with Continuous Functions

Let $f:{\mathcal{D}}_f\subseteq \mathbb{R}\to \mathbb{R}$ and $g:{\mathcal{D}}_g\subseteq \mathbb{R}\to \mathbb{R}$ be real functions and let $c\in \mathbb{R}$.

If $f$ and $g$ are continuous at $c$, then so are:

• $\alpha f+\beta g$ for all $\alpha ,\beta \in \mathbb{R}$;

• $f\cdot g$;

• $f/g$;

• $f\circ g$ provided that $g({\mathcal{D}}_g)\subseteq {\mathcal{D}}_f$.

PROOF

TODO

The Extreme Value Theorem

Let $f:\mathcal{D}\subseteq \mathbb{R}\to \mathbb{R}$ be a real function and let $[a;b]\subseteq \mathcal{D}$ be a closed interval.

If $f$ is continuous on $[a;b]$, then there exist at least one $x_{\text{of min}}\in [a;b]$ and at least one $x_{\text{of max}}\in [a;b]$ such that

$$f(x_{\text{of min}})\le f(x)\le f(x_{\text{of max}})\forall x\in [a;b]$$

INTUITION

This theorem says that if a function is continuous on a closed interval, then it has a minimum and a maximum value on it.

TODO: Add diagram

PROOF

TODO

The Intermediate Value Theorem

Let $f:\mathcal{D}\subseteq \mathbb{R}\to \mathbb{R}$ be a real function and let $[a;b]\subseteq \mathcal{D}$ be a closed interval.

If $f$ is continuous on $[a;b]$, then for every $y$ between $f(a)$ and $f(b)$, i.e. $\min \{f(a),f(b)\}\le y\le \max \{f(a),f(b)\}$, there exists at least one $x\in [a;b]$ such that $f(x)=y$.

INTUITION

The theorem says that if a function is continuous on a closed interval, then it must generate all values between its minimum and maximum value on said interval.

PROOF

TODO

Bolzano's Theorem

Let $f:\mathcal{D}\subseteq \mathbb{R}\to \mathbb{R}$ be a real function and let $[a;b]\subseteq \mathcal{D}$ be a closed interval.

If $f$ is continuous on $[a;b]$ and $f(a)<0$ and $f(b)>0$ (or vice-versa), then there exists at least one $x\in (a;b)$ such that $f(x)=0$.

PROOF

This is just a special case of the intermediate value theorem.

Theorem: Integrability of Continuous Functions

Let $f:\mathcal{D}\subseteq \mathbb{R}\to \mathbb{R}$ be a real function and let $[a;b]\subseteq \mathcal{D}$ be a closed interval.

If $f$ is continuous on $[a;b]$, then it is also Riemann-integrable on $[a;b]$.

PROOF

TODO