Suppose $ f(0) = 0 $ and $ |f'(x)| \leq |f(x)| $ for every $ x \in (0,1) $. Show that $ f(x) = 0 $ for all $ x \in [0,1] $.

Let $ f $ be continuous on $ [0,1] $ and differentiable on $ (0,1) $. Suppose $ f(0) = 0 $ and $ |f'(x)| \leq |f(x)| $ for every $ x \in (0,1) $. Show that $ f(x) = 0 $ for all $ x \in [0,1] $.

This problem is in the Lagrange and Rolle's theorem chapter. So far, I have established that there exists a point $ c \in (0,1) $ such that $ f(1) = f'(c) $, which implies $ |f(1)| \leq |f(c)| $. Intuitively, it seems that I could apply this argument iteratively and approach $ f(0) $ then show that if $ f(x) \neq 0$ the function isn't continuous. But I'm uncertain whether this is a sound approach or how to actually formalize it.