= Unbounded derivative obstruction to Riemann integrability
{title2=$f\text{ differentiable}\not\Longrightarrow f'\text{ Riemann integrable}$}
A proper <Riemann integral> on a compact interval requires a bounded integrand, but a <derivative> can be unbounded. For $f(x)=x^2\sin(1/x^2)$ away from zero and $f(0)=0$, the <difference quotient> gives $f'(0)=0$, whereas $f'(x)=2x\sin(1/x^2)-2\cos(1/x^2)/x$ for $x\ne0$. At $x_n=(2\pi n)^{-1/2}$, this derivative is $-2/x_n$ and is unbounded. The <fundamental theorem of calculus> version that integrates a derivative therefore needs an explicit integrability hypothesis.
Back to article page
