I am struggling to prove or disprove that the following function family is uniformly equicontinuous. $$F = \{f \in C^1[0,1]: \forall x \text{ } |f(x)| + \sqrt x |f'(x)| \leq 1 \}$$
First I tried to disprove the statement with functions like $f_n(x) = C\sqrt{x + \frac{1}{n}}$ (which are in $F$ for some real valued constant $C$) but I completely forgot that $\sqrt{x}$ is uniformly continuous so my counterexample failed.
Now I am beginning to think that this family is uniformly equicontinuous, but I can't prove it.
Well for $x, y$ such that $x < y$ and $|x - y| < \delta$ by mean value theorem I get $$|f(x) - f(y)| = |f'(c)|(y - x) \leq \frac{y - x}{\sqrt{c}} \leq \frac{\delta}{\sqrt{c}} \leq \frac{\delta}{\sqrt{x}}$$ for some $c \in [x, y]$. After that I am just out of ideas what to do next.
P.S. I want to use Arzelà–Ascoli theorem afterwards.