# Thread: If f is Riemann integrable and continuous, prove that F is differentiable

1. ## If f is Riemann integrable and continuous, prove that F is differentiable

Some help on this following question, please. Thanks a bunch!

Question: If $f$ is Riemann integrable on $[a,b]$ and continuous at $x_0$, prove that $F(x) = \int_{a}^{x} f(t) dt$ is differentiable at $x_0$ and $F'(x_0)=f(x_0)$. Show that if $f$ has a jump discontinuity at $x_0$, then $F$ is not differentiable at $x_0$.

2. I sure that you mean $F(x) = \int_a^x {f(t)dt}$.

Notice that $F(x_0 + h) - F(x_0 ) = \int_{x_0 }^{x_0 + h} {f(t)dt}$ and the length of that interval is $|h|$.

3. Consider the difference quotient: $\frac{F(x)-F(x_0)}{x-x_0} = \frac{\int_{a}^{x} f(t) \ dt- \int_{a}^{x_0} f(t) \ dt}{x-x_0} = \frac{1}{x-x_0} \int_{x_0}^{x} f(t) \ dt$.

Then you need to show that $\lim\limits_{x \to x_0} \frac{1}{x-x_0} \int_{x_0}^{x} f(t) \ dt = f(x_0)$.

Use the fact that $f(x_0) = \frac{1}{x-x_0} \int_{x_0}^{x} f(x_0) \ dt$. Also use the fact that $\left| \int_{a}^{b} f \right| \leq \int_{a}^{b} |f|$ provided that $f$ is Riemann integrable. You have to consider two cases: $x_0 < a$ and $x_0 > a$.