- Thread starter
- #1

- Apr 14, 2013

- 4,601

How can I show that the function

$$f=\left\{\begin{matrix}

0, \text{ if } x \in [0,1)\\

1, \text{ if } x \in (1,2]

\end{matrix}\right.$$

is continuous at $[0,1) \cup (1,2]$ using the definition of continuity?

A function $f:A \rightarrow \mathbb{R}$ is continuous at a point $x_0$:

$ \forall ε > 0$, $\exists δ > 0$ such that $\forall x \in A$ with

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

How can I use this to show the continuity at the whole interval?