Suppose I have a function $F:[0,1] \times [0,\infty) \rightarrow (0,1]$ of two variables $(s,t)$ that satisfies:
(1) $F(s,0)=1$ for each $s \in [0,1]$.
(2) $\lim_{t \rightarrow \infty} F(s,t)=0$ for all $s \in [0,1]$.
(3) For each fixed $s \in [0,1]$, the function $t\mapsto F(s,t)$ is strictly decreasing.
Consider the function $G:[0,1] \times [0,\infty) \rightarrow \mathbb{R}$ defined by $G(s,t)=\frac{\partial F}{\partial s}(s,t)$.
Is it necessarily the case that $\sup_{(s,t) \in [0,1] \times [0,\infty)} \left|G(s,t)\right| \lt \infty$?
If not, how would I go about constructing a counter-example?