Let $f_k, f \in C([0,1]; \mathbb{R})$ s.t. $f_k \to f$ uniformly as $k \to + \infty$. Suppose $t \in (0,1)$ and that $f$ is differentiable at $t$. Take $\tau_k$, a sequence of real numbers s.t. $\tau_k \to 0$ and s.t. $0\le t+\tau_k \le 1$ for every $k \in \mathbb{N}$.
Can we prove that
$$ \liminf_k \frac{f_k(t+\tau_k)-f_k(t)}{\tau_k} \ge f'(t) $$ ?