Let $f:X \to Y$ be a homotopy equivalence of pointed topological spaces.

Then, is the induced map of pointed loop spaces $\Omega (f): \Omega X \to \Omega Y$ a homotopy equivalence?

Here, loop spaces are equipped with the compact-open topologies.

Is there any counterexample?

I do not know even whether the induced map $\Omega (f)$ is continuous or not in general.