Is it possible to have a function differentiable but not continuous in a given interval?

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?

Solution 1:

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).

Solution 2:

No, this is not possible. However, you can have a function that is continuous but not differentiable (Weierstrass Function).