Continuity

Example 3.

1. The constant function \(f:\mathbb{R}\to\mathbb{R}\) with \(f(x)=c\) for some \(c\in\mathbb{R}\) is continuous on \(\mathbb{R}\).

2. The Heaviside function \[H(x)=\begin{cases}1&,\text{ if }x\geq 0,\\0&,\text{ if }x<0.\end{cases}\] is discontinuous at \(x_0=0\), but continuous everywhere else.

3. The function \(f:\mathbb{R}\to\mathbb{R}\) \[f(x)=\begin{cases}1&,\text{ if }x= 0,\\0&,\text{ if }x\neq0.\end{cases}\] is discontinuous at \(x_0=0\), but continuous everywhere else.

4. Polynomials are continuous on \(\mathbb{R}\).

5. Rational functions \(f:I\to\mathbb{K}\) with \(f(x)=\frac{p(x)}{q(x)}\) for some polynomials \(p,q\) (\(q\) is not the zero polynomial) are defined on \(I=\{x\in\mathbb{R}\,:\,q(x)\neq0\}\) and are continuous on \(I\) (due to the formulae for convergent sequences).

6. The absolute value function \(|\cdot|:\mathbb{R}\to\mathbb{R}\), i.e., \[|x|=\begin{cases}x&,\text{ if }x\geq 0,\\-x&,\text{ if }x<0.\end{cases}\] is continuous on \(\mathbb{R}\).

7. The function \(f:\mathbb{R}\to\mathbb{R}\) with \[f(x)=\begin{cases}1&,\text{ if }x\in\mathbb{Q},\\0&,\text{ if }x\notin\mathbb{Q}\end{cases}\] is everywhere discontinuous.
   Proof: Let \(x_0\in\mathbb{R}\):
   First case: \(x_0\in\mathbb{Q}\). Then take a sequence \((x_n)_{n\in\mathbb{N}}\) with \(\lim_{n\to \infty}x_n=x_0\) and \(x_n\in\mathbb{R}\backslash\mathbb{Q}\) (for instance, \(x_n=x_0+\frac{\sqrt{2}}{n}\)). Then \(f(x_n)=0\) for all \(n\in\mathbb{N}\) and thus \(\lim_{n\to \infty}x_n=0\neq f(x_0)=1\).
   Second case: \(x_0\in\mathbb{R}\backslash\mathbb{Q}\). Then take a sequence \((x_n)_{n\in\mathbb{N}}\) with \(\lim_{n\to \infty}x_n=x_0\) and \(x_n\in\mathbb{Q}\) (this exists since every real number can be approximated by a rational number in arbitrary good precision). Then \(f(x_n)=1\) for all \(n\in\mathbb{N}\) and thus \(\lim_{n\to \infty}x_n=1\neq f(x_0)=0\).