Is it true that $\exists a\in I\ \text{and}\ \epsilon>0$ such that $f$ is constant on $(a-\epsilon,a+\epsilon)\bigcap I$?

Question

Let $f:I\to\mathbb{R}$ be a simple function where $I$ is an interval with more than one point. Is it true that $\exists a\in I\ \text{and}\ \epsilon>0$ such that $f$ is constant on $(a-\epsilon,a+\epsilon)\bigcap I$? (Here, by a simple function is meant a function with finite range).

I think the answer to the question is YES. But I cannot find a way to prove this. I tried supposing the negation and tried to obtain a contradiction. Yet I went nowhere with my attempt. Can someone please give me a hint? Thanks.

Answer 1

Hint

What do you think about $\boldsymbol 1_{\mathbb Q\cap [0,1]}$ ?