is every differentiateable function integratable
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Well, If you are thinking Riemann integrable, Then every differentiable function is continuous and then integrable!
However any bounded function with discontinuity in a single point is integrable but of course it is not differentiable!
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Volterra's function is differentiable everywhere just as f (as defined above) is. One can show that f ′(x) = 2x sin(1/x) - cos(1/x) for x ≠ 0, which means that in any neighborhood of zero, there are points where f ′ takes values 1 and −1.
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