Sign preservation for a function of two variables

Let $V\subset R^n$ and let $f\colon V\times(0,\infty)\to R$ be a continuous function. In the article we study the properties of $f(x,y)$ by means of one dimensional functions $f_x(y)=f(x,y)$, with $x\in V$ fixed. We determine the conditions on the functions $f_x(y)$ which are necessary and sufficient for $f$ to have constant sign in a neigborhood of the point $(x,0)$ for some $x\in V$. We also state different conditions $\{f_x\}_{x\in V}$ which are sufficient for $f$ to be indentically zero. This is of use in proving unicity of solution for a number of problems of mathematical physics, in solving inverse problems and in studying questions of stability for solutions to equations.