- #1
Niles
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Homework Statement
We have a piecewise continuous function and T-periodic function f and we have that:
[tex]
F(a) = \int_a^{a + T} {f(x)dx}
[/tex]
I have to show that F is diferentiable at a if f is continuous at a.
My attempt so far:
I have showed that F is continuous for all a. If we look at one piece where f is continuous for a, we have that if F'(a) = f(a):
[tex]
\frac{1}{h}\int_a^{a + h} {f(x)dx = \frac{{\int_x^{a + h} {f(x)dx - } \int_x^a {f(x)dx} }}{h}}
[/tex]
When h -> 0, then the above goes to f(a) - this is just the Fundemental Theorem of Calculus.
Now I have that:
[tex]
\frac{{F(a + h) - F(a)}}{h} = \frac{{\int_{a + h}^{a + h + T} {f(x)dx - } \int_a^{a + T} {f(x)dx} }}{h} = \frac{{\int_{a + T}^{a + h + T} {f(x)dx - } \int_a^{a + h} {f(x)dx} }}{h}
[/tex]
But where to go from here?