Let $f: [0, 1] \to \mathbb R$ be an $L^1$ function. Define for each $r > 0$, the blow up $f_r:[0, 1] \to \mathbb R$ by

$$f_r (x) := \frac{f(rx)}{r}.$$

Suppose $f_r$ converges in $L^1$ to some function $g$ as $r \to 0^+$. Is it true that $g$ agrees a.e. with a linear function?