# Cauchy definition (epsilon-delta) of continuous functions

• Nov 28th 2009, 03:00 AM
SubZero
Cauchy definition (epsilon-delta) of continuous functions

Attachment 14092

What i Know

I know that this graph is not continuous at a=0 but i am finding it difficult to prove this using Cauchy definition (epsilon-delta) of continuous functions.

Thanks Subzero
• Nov 28th 2009, 03:56 AM
HallsofIvy
Quote:

Originally Posted by SubZero

Attachment 14092

What i Know

I know that this graph is not continuous at a=0 but i am finding it difficult to prove this using Cauchy definition (epsilon-delta) of continuous functions.

Thanks Subzero

Good! You are probably finding it difficult to show that function is not continous at 0 because it is continuous at 0!

Given any $\displaystyle \epsilon> 0$ if x> 0, f(x)= x so we can just take $\displaystyle \delta= \epsilon$. That way, "if $\displaystyle |x- 0|= |x|> \delta$, $\displaystyle |f(x)- f(0)|= |x- 0|= |x|< \delta= \epsilon$". While if x< 0, f(x)= 0 so $\displaystyle |f(x)- f(0)|= |0- 0|= 0< \epsilon$ for any x and so for $\displaystyle |x- 0|= |x|< \delta$.

(The function is not differentiable at x= 0 but is continuous there.)
• Nov 28th 2009, 04:11 AM
SubZero
Thanks for the help