Tag Archives: Continuous

εδ Confusion in Limit & Continuity

Modern Math 2 Comments

1. Basic:
|y|= 0 or > 0 for all y

2. Limit: \displaystyle\lim_{x\to a}f(x) = L ; x≠a
|x-a|≠0 and always >0
hence
\displaystyle\lim_{x\to a}f(x) = L
\iff
For all ε >0, there exists δ >0 such that
\boxed{0<|x-a|<\delta}
\implies |f(x)-L|< \epsilon

3. Continuity: f(x) continuous at x=a
Case x=a: |x-a|=0
=> |f(a)-f(a)|= 0 <ε (automatically)
So by default we can remove (x=a) case.

Also from 1) it is understood: |x-a|>0
Hence suffice to write only:
|x-a|<\delta

f(x) is continuous at point x = a
\iff
For all ε >0, there exists δ >0 such that
\boxed{|x-a|<\delta}
\implies |f(x)-f(a)|< \epsilon

Differential Theorem

Elementary Math, Modern Math 1 Comment

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