Prove continuity by first principle

1. Oct 5, 2011

chocolatefrog

1. The problem statement, all variables and given/known data

Prove that f(x) = x^2 is continuous at x = 2 using the ε - ∂ definition of continuity.

2. The attempt at a solution

Using the definition of continuity, I've reached thus far in the question:

|x - 2||x + 2| < ε whenever |x - 2| < ∂

3. Relevant equations

I have no clue how to move forward from here. I know while solving this type of questions, we try to solve the first inequality so that ∂ can be written in terms of ε, but I can't seem to figure out what to do with |x + 2|.

2. Oct 5, 2011

HallsofIvy

Staff Emeritus
Okay, $|x-2||x+2|< \epsilon$ and so $|x-2|< \epsilon/|x+ 2|$. That is almost what you want- you just need to replace that |x+2| with a constant. To do that, take some reasonable restriction on |x- 2|.

4. Oct 5, 2011

chocolatefrog

Oh, and I totally forgot, thanks a lot for your help! :)