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 $\epsilon> 0$ there exist $\delta_1> 0$ such that if $|x- a|< \delta_1$ then $|f(x)- f(a)|< \epsilon/2$ and there exist $\delta_2> 0$ such that if $|y- a|< \delta_2$ then $|f(y)- f(a)|< \epsilon/2$.

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