Compare $L^\infty$ norms of a function

Question

Let $D\subset \mathbb{R}^n$ bounded domain and $f: [0,t_0] \times D\longmapsto \mathbb{R}$ a function. Is that $$\|f(0,\cdot)\|_{L^\infty(D)} \leq \|f\|_{L^\infty((0,t_0)\times D)}.$$ The problem is for set of zero measure. Maybe this inequality or similar one holds with a constant.

Answer 1

As pointed out by Arctic Char the inequality fails. Even if $(0,t_0)$ is replaced by $[0,t_0]$ on the right side it is still false. This is because RHS does not depend on the values of $f(0,x)$ in view of the fact that $\{0\} \times \mathbb R^{n}$ has measure $0$. Take $f(t,x)=0$ for $t \neq 0$. In this case RHS is $0$. But LHS can be any number.