Proving the continuity of a function

Question

Prove that $f\colon \mathbb R^2 \to \mathbb R$, $(x, y) \to ye^x + \sin x + (xy)^4$ is continuous.

Let $y = x, x = 0$, then $$\lim_{(x, y) \to (0,0)} ye^x + \sin x + xy^4 = 0.$$

I have proved this is continuous at $(0,0)$, but how do I prove it is continuous everywhere?

Answer 1

It is a sum of continuous functions, so it is continuous everywhere.