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

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

https://www.desmos.com/calculator/5syacx64uy

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