Continuity of Real Functions

Continuity

Definition: Continuity of Real Functions

A real function $f:D\subseteq \mathbb{R}\to \mathbb{R}$ is continuous at $c\in D$ if its limit as $x$ approaches $c$ is exists and is equal to its value for $c$:

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

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

Properties

Theorem: Operations with Continuous Functions

If $f,g:D\subseteq \mathbb{R}\to \mathbb{R}$ are two continuous functions, then so are

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

• $f\cdot g$;

• $f/g$ provided that $g(x)\ne 0$ for all $x\in D$;

• $f\circ g$ provided that $g(D)\subseteq D$.

PROOF

TODO

The Extreme Value Theorem

Let $f:D\subseteq \mathbb{R}\to \mathbb{R}$ be a real-valued function on a closed interval $D=[a;b]$.

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

$$f(x_{\text{of min}})\le f(x)\le f(x_{\text{of max}})\forall x\in D$$

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 $f:D\subset \mathbb{R}\to \mathbb{R}$ be a real function on a closed interval $D=[a;b]$.

If $f$ is continuous on $D$, 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 an $x\in D$ 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 the values on the interval’s endpoints.

PROOF

TODO

Bolzano's Theorem

Let $f:D\subset \mathbb{R}\to \mathbb{R}$ be a real function on a closed interval $D=[a;b]$.

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

PROOF

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

Theorem: Integrability of Continuous Functions

Let $f$ be a real function on a closed interval.

If $f$ is continuous, then it is also Riemann-integrable.

PROOF

TODO