Continuity of $t\mapsto f(t,h(t))$

Question

Let $f:[0,T]\times X \to Y$ be a map where $X$ and $Y$ are Hilbert spaces. We have that $t \mapsto f(t,x)$ is continuous, and so is $x \mapsto f(t,x)$.

Let $h:[0,T] \to X$ be continuous. Does it follow that $$t \mapsto f(t, h(t))\quad\text{ is continuous}?$$

$f$ is better than a Caratheodory function, so I am hoping it holds. Unfortunately the literature is mainly devoted to purely Caratheodory functions (where the $t$ argument is only measurable) so I couldn't find this information. I know this is like a "composition of continuous functions is continuous" thing but there are two arguments here.

Answer 1

Look at this question and then create $h$ such that it moves to and fro the two paths used in the answer to provide a counter example: