Real Analysis Continuity problem.

[quantchem]

Homework Statement

Show that |f(x) - f(y)| < |x - y| if f(x) = sqrt(4+x^2) if x is not equal to x_o. What does this prove about f?

Homework Equations

The Attempt at a Solution

Already proved the first part. I am guessing that for the second part the answer is that f is continuous but I am not really sure how to show it. Please help.

 

[Dick]

Well, sure. It's continuous. What's the definition of continuity?

 

[quantchem]

I did this :

let
|f(x) - f(y)| < epsilon and |x - y| < delta. then epsilon < delta and therefore for each epsilon > 0, there is some delta > 0 such that |f(x) - f(y)| < epsilon when |x - y| < delta. so f is continuous at y when x is in an interval where f is defined.

Is this sufficient explanation?

 

[Dick]

I did this :

let
|f(x) - f(y)| < epsilon and |x - y| < delta. then epsilon < delta and therefore for each epsilon > 0, there is some delta > 0 such that |f(x) - f(y)| < epsilon when |x - y| < delta. so f is continuous at y when x is in an interval where f is defined.

Is this sufficient explanation?

Sure. So given any epsilon, you can pick delta=epsilon and satisfy the requirements of continuity, right?

 

[quantchem]

yup. thanks.