Suppose $f(t,x)\in \mathcal{C}^0([0,1]\times \mathbb{R}^n)$. Further suppose that for each $t$
$$ C(t):= \sup\_{x\in\mathbb{R}^n} |f(t,x)|<\infty \, .$$


Does it follow that $f$ is bounded? 


Note that $C(t)$ need not be continuous as is evidenced by $f(t,x)= \sin(t x)$.



Extending point-wise bound to uniform bound
It does not. Start from any non-zero continuous function $\phi$ with compact support in $\mathbb{R}\setminus\{0\}$ and define 


$f(t,x)=\phi(tx)/t$ for $t>0$ and $f(0,x)=0$.


