Proving f(z) is a continuous function in the entire complex plane

[lonewolf999]

Homework Statement

Show that the function f(z) = Re(z) + Im(z) is continuous in the entire complex
plane.

Homework Equations

The Attempt at a Solution

I know that to prove f(z) is a continuous function i have to show that it is continuous at each part of its domain.

I take it that means i have to prove that Re(z) and Im(z) are continuous, however i have tried reading through my notes on how to do this and havn't been able to come up with a starting point.

 

[Coto]

Try writing out z as the usual combination of x + iy. Then what is Re(z) and Im(z)? Are these functions continuous?

 

[lonewolf999]

if i take z=x+iy, then Re(z)=x and Im(z)=y, however i can't finish it there.

 

[Coto]

Is the function f(z) = x continuous? Is the function f(z) = y continuous? Start with the definition of continuity.

 

[arkajad]

Is it by some chance true that $$|Re(z)|\leq |z|$$?