Continuous in two sets implies continuity in union of sets

Question

Say $f(x) = 0, x\in[0,1) $ and $f(x) = 1, x\in (1,2] $. Show that $f(x)$ continuous in $[0,1) \cup (1,2]$.

I can see that $f(x)$ is continuous in $[0,1)$ and $(1,2]$, but what does that imply that it's continuous in the union?

Answer 1

Hint: If $x_0\in[0,1)\cup(1,2]$ and if $\delta>0$ is so small that:

• if $x_0\in[0,1)$, $(x_0-\delta,x_0+\delta)\subset[0,1)$;
• if $x_0\in[0,1)$, $(x_0-\delta,x_0+\delta)\subset(1,2]$.

Then, $|x-x_0|<\delta\implies|f(x)-f(x_0)|=0$.