In my single variable calculus book the derivation of the fundamental theorem of calculus only covered a special case and I would like help with generalisering the proof.
In my book the following was proven: If a function $f$ is continuous and differentiable on $(a, b)$ then it may be integrated over $[\alpha, \beta]$ satisfying $[\alpha, \beta] \subset (a,b)$ and its value is given by $F(\beta) - F(\alpha)$.
However, I believe the general case allows for $f$ to be integrated over $[a, b]$ given that $f$ is also continuous on $[a, b]$. This case was not proven. So I suppose that if i let $\alpha \to a^+$ and $\beta \to b^-$ and if I somehow show that the continuity of $f$ implies the continuity of $F$ on $[a, b]$ then perhaps that is enough to show that the integral of $f$ over $[a, b]$ is given by $F(b) - F(a)$ as well. How do I prove this?