Fundamental theorem of calculus I - Imported Knowlpedia Content

Fundamental theorem of calculus I: Let $a<b$ and let $f:[a,b]\to\mathbb{R}$ be a Riemann integrable function. Define

F(x)=\int_a^x f(t)\,dt \quad (x\in[a,b]).

Then $F$ is continuous on $[a,b]$ . Moreover, if $f$ is continuous at a point $c\in(a,b)$ , then $F$ is differentiable at $c$ and

F'(c)=f(c).

Remarks

This links the Riemann integral to the derivative by showing the integral produces an antiderivative wherever the integrand has no discontinuity.