Continuous Function

analysis

Definition

Continuous Function

A function $f:D\to \mathbb{R}$ is called continuous at $x_0\in D$ if the limit of the function as $x$ approaches $x_0$ exists and is equal to the function value at that point:

$$f(x_0)=\underset{x\to x_0}{\lim }f(x)$$

The function $f$ is called continuous on $D$ if it is continuous for all $x_0\in D$.

Epsilon-Delta Definition

Alternatively, $f$ is continuous at $x_0$ if for every $\epsilon >0$, there exists a $\delta >0$ such that for all $x\in D$:

$$|x-x_0|<\delta ⟹|f(x)-f(x_0)|<\epsilon$$

Sequential Continuity

Continuity is equivalent to sequential continuity: $f$ is continuous if and only if for every sequence $(x_n{)}_{n\ge 0}\subseteq D$:

$$\underset{n\to \infty }{\lim }{x}_n=a⟹\underset{n\to \infty }{\lim }f(x_n)=f(a)$$

Properties

Existence of a Delta Neighbourhood

Existence of a Delta Neighbourhood

For every continuous function $f$ with $f(x_0)>0$, there exists a delta neighbourhood $U_{\delta }(x_0)$ such that $f(x)>0$ for all $x\in U_{\delta }(x_0)$. Analogously, the same applies for $f(x_0)<0$.

Existence of a Delta Neighbourhood

Let $f$ be continuous and $\epsilon =\frac{|f(x_0)|}{2}>0$. By the definition of continuity, there exists a $\delta >0$ such that for all $|x-x_0|<\delta$:

$$|f(x)-f(x_0)|<\frac{|f(x_0)|}{2}$$

For $f(x_0)>0$, this implies $f(x)>f(x_0)-\frac{f(x_0)}{2}=\frac{f(x_0)}{2}>0$.

Preservation of Sign

For every continuous function $f$ with $f(x_0)>0$, there exists a delta neighbourhood $U_{\delta }(x_0)$ such that $f(x)>0$ for all $x\in U_{\delta }(x_0)$. For the case $f(x_0)<0$, an analogous statement holds.

Image of a Closed Interval

Let $I\subseteq \mathbb{R}$ be a closed interval and $f:I\to \mathbb{R}$ a continuous function. Then $f(I)$ is also a closed interval. (See Extreme Value Theorem).

Interval on Strictly Monotone Continuous Function

Let $I=[a,b]$ be an interval and $f:I\to \mathbb{R}$ be a strictly monotonic continuous function. Then there exists an inverse function $f^{-1}:f(I)\to I$ that is also continuous.