I've been studying different types of functions and I came across one on What is an example that a function is differentiable but derivative is not Riemann integrable, but I can't figure out why $f(x)=x^{ \frac{3}{2} }sin(\frac{1}{x})$ on $[0,1]$ is continuous, because it seems that that it doesn't exist at $x=0.$ But I know it is differentiable on $(0,1),$ and not Riemann integrable. Some clarification, please?
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