If $f$ is a continuous function then Argmax of $f$ is continuous?

Question

Let $I=[a,b]$ a closed real interval

Let $f: I \to R$ be a continuous function

Let $y : I \to I$ be a function such that $\forall x \in I:$ $$ \sup_{t \in [a,x]}f(t) = f(y(x)) $$ I would like to know if $y$ is a continuous function.

Thanks

Answer 1

Think of $\sin(x)$ on $[\pi,3\pi]$. The argmax starts out at $\pi$ and stays there until $x\geq 2\pi$ at which point it jumps to be greater than $2\pi$.

Answer 2

Take $f(t) = 1$ for all $t \in I$ and choose an arbitrary discontinuous function $y : I \to I$.

This rejects your claim!