Continuity proof, not sure how to put it together.

[math-help-me]

Homework Statement

Prove that if f is continuous at a, then for any ε>0 there is a σ>0,?
such that if abs(x-a)< σ and abs(y-a)< σ then abs[f(x) - f(y)]< ε

Homework Equations

Definition of continuity and triangle inequality
abs(f(x)-f(y))= abs(f(x)-f(a) + f(a)-f(y))≤ abs(f(x)-f(a))+ abs(f(y)-f(a))

The Attempt at a Solution

So i think I need to apply the continuity definition twice and bring things together with the triangle inequality but i don't know how to go about it all.

 

[HallsofIvy]

Yes, that's exactly what you need. Since f is continuous at a, given any \epsilon&gt; 0 there exist \delta_1&gt; 0 such that if |x- a|&lt; \delta_1 then |f(x)- f(a)|&lt; \epsilon/2 and there exist \delta_2&gt; 0 such that if |y- a|&lt; \delta_2 then |f(y)- f(a)|&lt; \epsilon/2.

Take \delta= min(\delta_1, \delta_2) so that if |x- a|&lt; \delta, both are true.