Let $f$ be a real-valued function on a closed interval $[a,b]$ undefined only at finite points in $(a,b)$. Let $F$ be antiderivative of $f$. Then:
$$\int_a^b f(x)\ dx=F(b)-F(a)$$
Is the theorem true? How shall we prove it?
I ask this because I have read that function undefined at finite points in the interval $(a,b)$ doesn't effect integration i.e. we can take the antiderivatives and apply the limits as usual to get the definite integral.