Find a continuous function $f(x)$ such that $f(x) \leq x$ and $0 \leq f(x) \leq C$ and $f(0)=0$ and $f'(x) > 0$

As the title says, I am looking for a continuous (integrable) function that is bounded between $0$, $x$, and a constant $C$ that is monotonically increasing. All conditions (except continuity) only need apply for $x \geq 0$. Ideally the slope of the function is close to $x$ but that is optional.

Here is a desmos illustrating the problem with a partial solution that is not bounded by $x$ for all $C$.

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