I have problems solving the following task:
I have the function $f(x)=\sin\frac{1}{x}$ when x isn't $0$ and $f(0) = 1$
First, I must prove that $f(x)$ is integrable in every interval $[a, b]$.
Second, I have $F(x) = \int_0^x f(t)dt$. I must find the derivative of $F(x)$.
What I've done so far is this: $f(x)$ is integrable over every $[a,b]$ which doesn't include $0$ and for $0$ I take a Riemman sum with $0$ in the interval. There I just split the interval and take the normal Riemman sums $f(c_i) * \Delta x_i$ and take $c_i$ to be $f(0)$ when $\Delta x_i$ has $0$. Obviously if I take the $lim_{\Delta x_i \rightarrow 0} f(c_i) \Delta x_i$ it exists and therefore there is an integral. Is this correct?
For the second part I've managed to prove that $F(x)$ is continous at $0$ but can't prove that it's differentiable there. Everywhere else $F'(x) = f(x)$ from the Leibniz-Newton theorem.
Can you please help me?