non analytic functions

Question

Find two functions, each of which is nowhere analytic, but whose sum is an entire function.

I can give examples of functions that are analytic nowhere, but can't find two that add to an entire function.

Answer 1

$f_1(z) = \operatorname{re} z$, $f_2(z) = i\operatorname{im} z$.

Then $f_1(z)+f_2(z) = z$.

Answer 2

Silly example: let $f: \mathbb{C} \to \mathbb{C} $ be nowhere analytic. Then if $g: \mathbb{C} \to \mathbb{C} $ is analytic, $g(z)-f(z)$ is also nowhere analytic. But $$ f(z) + (g(z)-f(z)) = g(z), $$ which is analytic.