- #1

dillingertaco

- 12

- 0

## Homework Statement

Prove the function f(x)= { 4 if x=0; x^2 otherwise

is discontinuous at 0 using epsilon delta.

## Homework Equations

definiton of discontinuity in this case:

there exists an e>0 such that for all d>0 if |x-0|<d, |x^2-4|>e

## The Attempt at a Solution

I'm confused because if we include ALL delta >0, eventually (namely, x around +/- 2) |x^2-4| will be less than e for all e>0 which seems to me to point to it being continuous at 0 when it clearly is NOT. Is there something built into the definition that ignores large values of delta which makes the interval around x too large?

So my way:

Assume it is continuous at 0. Let e=1

Then |x^-4|<1 when |x|<d for some d.

From here I want to say d<1 and so |x^2-4|>2 for all |x|<d which would be the contradiction but I don't think that's how I formally say it...