If a function is discontinuous, why doesn't it contradict the Fundamental Theorem of Calculus

Question

So for example, suppose $g(x) = \int_o^x F(t) dt$ where $$F(t) = \begin{cases} t & 0 \leq t \leq 1 \\ t - 1 & 1 < t \leq 2 \end{cases} $$

The function $g$ is not differentiable at 1. I am curious to why it doesn't contradict the Fundamental Theorem of Calculus. Some thoughts and assistance

Answer 1

Because the Fundamental Theorem of Calculus states that if $f$ is continuous, then$$x\mapsto\int_a^xf(t)\,\mathrm dt$$is differentiable. And your function $F$ is not continuous.