Is there any possible function that is not continuous but differentiable in a given interval. It sounds non-logical to me since differentiation is a special limit function in itself therefore non-continuous should be meaning non-differentiable either. Am I right?
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$\begingroup$No: A function which is differentiable at $x$ is continuous at $x$. To prove this, note that the quantity
$$\left|\frac{f(x) - f(y)}{x - y}\right|$$
is a bounded function of $y$ in a deleted neighborhood containing $x$; if $M$ is a bound, then rearranging shows that
$$|f(x) - f(y)| \le M |x - y|$$
for all $y$ in the deleted neighborhood (and in fact, in the neighborhood).
$\endgroup$ $\begingroup$No, this is not possible. However, you can have a function that is continuous but not differentiable (Weierstrass Function).
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