Continuity and Differentiability of Step function?

Question

All Differentiable functions must be continous , But step function is differentiable and its derrivative is Dirac delta function, Step function actually is not continous But it have Derrivative , How is this possible??

Answer 1

My guess is that you are calling “step function” to a function like$$\begin{array}{rccc}s\colon&\mathbb R&\longrightarrow&\mathbb R\\&x&\mapsto&\begin{cases}0&\text{ if }x<0\\1&\text{ otherwise.}\end{cases}\end{array}$$Yes, it is a function and, yes, it is not continuous (at $0$). But it is not differentiable (again, at $0$). It is oftain said that $s'$ is the Dirac function, but that is only an idea about what the Dirac function is. Actually, the limit $\displaystyle\lim_{x\to0}\frac{s(x)-s(0)}x$ doesn't exist (in $\mathbb R$) and therefore $s$ is not differentiable at $0$.

Answer 2

Your initial thesis is incorrect: step functions are not differentiable at the points of discontinuity, and the Dirac Delta is not a function.