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

1. The problem statement, all variables and given/known data
Show that the function f(z) = Re(z) + Im(z) is continuous in the entire complex

2. Relevant equations

3. 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.
Try writing out z as the usual combination of x + iy. Then what is Re(z) and Im(z)? Are these functions continuous?
if i take z=x+iy, then Re(z)=x and Im(z)=y, however i cant finish it there.
Is the function f(z) = x continuous? Is the function f(z) = y continuous? Start with the definition of continuity.
Is it by some chance true that [tex]|Re(z)|\leq |z|[/tex]?

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