Continuous Function

analysis

Definition

Continuous Function

A function $f:D\to \mathbb{R}$ is called continuous in $x_0\in D$ if $f(x_0)={\lim }_{x\to \infty }f(x)$. The function $f$ is called continuous in $D$ if $f$ is continuous for all $x_0\in D$.

A function $f:D\to \mathbb{R}$ is called continuous in $x_0\in D$ if:

$$\forall \epsilon >0:\exists \delta (\epsilon )>0:\forall x:\left(|x-x_0|<\delta (\epsilon )⟹|f(x)-f(x_0)|<\epsilon \right)$$

For continuous functions, the following holds:

$$f\text{ continuous}⟹\underset{n\to \infty }{\lim }f(x_n)=f\left(\underset{n\to \infty }{\lim }{x}_n\right)$$

since continuity is equivalent to sequential continuity:

$$f\text{ sequentially continuous}:⟺\forall (x_n{)}_{n\ge 0}:\underset{n\to \infty }{\lim }{x}_n=a⟹\underset{n\to \infty }{\lim }f(x_n)=f(a)$$

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 a continuous function and $\epsilon =\frac{f(x_0)}{2}$. Due to the continuity of $f$, there exists a $\delta$ such that

$$f(x)>f(x_0)-\epsilon =\frac{f(x_0)}{2}>0$$

for $|x-x_0|<\delta$.

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.

Closed Interval

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

Interval on Strictly Monotone Continuous Function

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