Differential Theorem

If f **differentiable** at point a => f **continuous** at point a

Converse not true !

‘Differentiability’ stronger than ‘Continuity’**Are all Continuous functions Differentiable ? False!****Counter-example (by Weierstrass)**:

$Latex f(x)=Sigma{b^n}cos(a^npi x)$ n ∈[0,8], a= odd number, b∈[0,1], ab > 1+3Π/2

f(x) Continuous everywhere (cosine), but non-differentiable everywhere!

Note: Weierstrass Function is the first known fractal. (e.g. Snowflake Koch’s curve).

Plot of Weierstrass Function (Photo credit: Wikipedia)

Note: What it means a curve (function) is :

1. Continuous = not broken curve

2. Differentiable = no pointed ‘V‘ or ‘W‘ shape curve