Ian Charlesworth

Suppose $f : [a, b] \to \R$ is integrable, and define $F : [a, b] \to \R$ by $$F(t) = \int_a^t f(x)\,dx.$$ Then $F$ is continuous at $t_0 \in (a, b)$...

1. ...always.
2. ...provided that there is some $\epsilon \gt 0$ so that $f$ is continuous on $(t_0-\epsilon, t_0+\epsilon)$.
3. ...provided that $f$ is continuous at $t_0$.
4. ...provided that $f$ is differentiable at $t_0$.

Suppose $f : [a, b] \to \R$ is integrable, and define $F : [a, b] \to \R$ by $$F(t) = \int_a^t f(x)\,dx.$$ Then $F$ is differentiable at $t_0 \in (a, b)$...

1. ...always.
2. ...provided that there is some $\epsilon \gt 0$ so that $f$ is continuous on $(t_0-\epsilon, t_0+\epsilon)$.
3. ...provided that $f$ is continuous at $t_0$.
4. ...provided that $f$ is differentiable at $t_0$.