Differential Theorem

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):

f(x)=\Sigma{b^n}cos(a^n\pi 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

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

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