If at optimum the objective function is monotonic in the parameter, do partial derivative with respect to the parameter exist almost everywhere?

Question

Let $X$ denote an arbitrary choice set, and let $t\in[0,1]$ be the relevant parameter.

Let $f:X\times[0,1]\rightarrow R$ be the parameterized objective function.

Let $V(t)=\sup_{x\in X}f(x,t)$ be the value function.

Let $X^*(t)=\{x\in X:f(x,t)=V(t)\}$ be the set of maximizers.

Suppose that for every $x\in X$, $f_t(x,t)$ exists almost everywhere. Suppose that for any selection $x^*(t)\in X^*(t)$, $f(x^*(t),t)$ is monotone in $t$. Is it true that $f_t(x^*(t),t)$ exists for almost all $t\in[0,1]$?

I know that a monotonic function is differentiable almost everywhere. But here I am looking at partial derivative. So I am not sure if the claim is true.