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

Does it follow that $f$ is bounded?

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

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 $f(t,x)= \sin(t x)$.

Suppose $f(t,x)\in \mathcal{C}_0([0,1]\times \mathbb{R}^n)$. Further suppose that for each $t$ $$ C(t):= \text{sup}_{x\in\mathbb{R}^n} |f(t,x)|<\infty $$.

Does it follow that $f$ is bounded?

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

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

Does it follow that $f$ is bounded?

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