Let $f:ℝ²→ℝ$ be an arbitrary harmonic function. **A level curve in two dimensions is a curve on which the value of a function $f(x,y)$ is a constant**. My **question** is: Find sufficient and necessary conditions on $f$ in which the level curve $f(x,y)=0$ implies only one case $x=a$ for all real $y$ (this mean that there exists a unique real number $a$ such that for all $y$, $f(x,y)=0$ implies $x=a$). The same question for complex functions from the complex plane into itself. 



Find sufficient and necessary conditions on $f$ in which the level curve $f(x,y)=0$ implies only one case $x=a$ for all real $y$
Modulo an elementary transformation you are asking for harmonic functions which are zero on a line and nowhere else. Such a function is linear. This is Theorem I in [*Short proofs of three theorems on harmonic functions*](http://www.ams.org/journals/proc/1988-102-04/S0002-9939-1988-0934865-6) by H.P. Boas and R.P. Boas
(published in Proc. Amer. Math. Soc. 102 (1988), 906-908).


