A question about a function that is uniformly continuous and nonzero on an interval

Question

I got this question:

Let $f$ be a function defined on an interval $I$ and let $0<L$ be a constant,
If $f$ is uniformly continuous on $I$ and $\forall x\in I, L \leq f(x)$, Must it be the case that $1/f(x)$ also uniformly continuous on $I$?

I think this statement to be false but I didn't managed to find a counter example. Thanks.