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
Singapore Maths Tuition 