# Thread: Cauchy definition (epsilon-delta) of continuous functions

1. ## Cauchy definition (epsilon-delta) of continuous functions

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

2. Originally Posted by SubZero

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

(The function is not differentiable at x= 0 but is continuous there.)

3. Thanks for the help