The solvability of a system of nonlinear equations

It is proved: if $\phi(\tau,\xi)$ is scalar continuous real function of arguments $\tau\in [a_{(n-1)},\ b_{(n-1)}]\subset R^{n-1},\ \xi\in [a,\ b]\subset R^{1}$ and $\phi(\tau,a) \phi(\tau,b)<0\ \forall \tau, $ then for each $\varepsilon >0$ exists a continuous $\phi_{0}(\tau,\xi),$ that $|\phi(\tau,\xi)- \phi_{0}(\tau,\xi)|<\varepsilon $ and the equation $\phi_{0}(\tau,\xi)=0$ has continuously depends on $\tau$ solution. The statement is suitable to a proof of a solvability finite system nonlinearity equations, to an estimation of a number of solutions. We give illustrating examples.