If $u(x, y)$ and $v(x, y)$ are continuous (respectively differentiable) does it follow that $f(z) =u(x, y) + iv(x, y)$ is continuous (resp. differentiable)? If not, provide a counterexample.

This came up in a Complex Analysis book I've been reading, and it seems to me that the answer is obviously yes by Cauchy Riemann, but it is extremely unlike this text to make anything that straightforward, so I suspect I am missing something.

Thanks in advance for the help.