real analysis - Continuity & boundedness on open interval implies uniform continuity

Suppose $f(x)$ is continuous and bounded on $(0,1)$. Is $f(x)$ uniformly continuous on $(0,1)$?

I think yes, because it's bounded, i.e. there exists $M: |f(x)| < M$. We could use this M as $\delta$ in the definition of uniformly continuous function for any $\epsilon$. My textbook says, the answer is no. Why?