Let
be fixed. The
difference quotient
is
-
![{\displaystyle {}{\frac {F(x+h)-F(x)}{h}}={\frac {1}{h}}{\left(\int _{a}^{x+h}f(t)\,dt-\int _{a}^{x}f(t)\,dt\right)}={\frac {1}{h}}\int _{x}^{x+h}f(t)\,dt\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2d080b122054dac6221f95aab430e48e3d1dd208)
We have to show that for
, the
limit
exists and equals
. Because of
the Mean value theorem for definite integrals,
for every
, there exists a
with
-
![{\displaystyle {}f(c_{h})\cdot h=\int _{x}^{x+h}f(t)dt\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12faa08d46a14f9201a9050dce5bca060087bb02)
and therefore
-
![{\displaystyle {}f(c_{h})={\frac {\int _{x}^{x+h}f(t)dt}{h}}\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9aac5c0160a18700b62de640436d98165ac4559b)
For
,
converges to
, and because of the continuity of
, also
converges to
.