Uniform estimate for function of two variables. A counterexample?

Question

Let $f:\mathbb{R}_+\times\mathbb{R}\to\mathbb{R}_+$, $\mathbb{R}_+:=[0,+\infty)$. Next, let $f(\cdot,x)\in C^1(\mathbb{R}_+)$, for all $x\in\mathbb{R}$, and $f(t,\cdot)\in C(\mathbb{R})$, for all $t\in\mathbb{R}_+$. Let $A\subset \mathbb{R}$ be bounded, and suppose there exists $a>0$, $b>0$, such that $$ f(0,x)>a, \quad f'_t(0,x)>b, \quad \forall x\in A. $$ Whether there exists $T=T(A)>0$ such that $f(t,x)>a$, for all $t\in[0,T]$, $x\in A$? My feeling is that this $T$ couldn't be, in general, uniform in $x$, however, I can't produce a counterexample.