Let $f(z) = u(x,y) + iv(x,y)$. I need to show that $f(z)$ is continuous if both $u(x,y)$ and $v(x,y)$ are continuous and differentiable if both $u(x,y)$ and $v(x,y)$ are differentiable.
What's the most efficient way to do the problem? Is there a way around using the limit definition?
You can replace the "if" with an "iff". In the final analysis the various claims follow from the fact that for all $i\in[n]$ the triangle inequality gives $$|x_i-y_i|\leq|x-y|\leq\sum_{k=1}^n|x_k-y_k|\ .$$ This implies that for any vector-valued function $t\mapsto {\bf f}(t)$ one has $$\lim_{t\to t_0}{\bf f}(t)={\bf a}\qquad\Leftrightarrow\qquad \lim_{t\to t_0}f_k(t)=a_k\quad(1\leq k\leq n)\ ,$$ whatever the intended limit $t\to t_0$ may be.
