# Proof of theorem about saving sign of continous function at point

Theorem : Let $M_0=(x_1,...,x_m) \in D(f) \subset R^m$ be the point of continuity of a function $f(M)$. If $f(M_0)\neq 0$ and $f(M_0)>0$, then $\exists B_{\delta}(M_0) \subset D(f)$ open ball with radius $\delta$, s.t. $\forall M \in D(f) \cap B_\delta(M_0)$ holds $f(M)f(M_0)>0$.

I understand the meaning of the theorem, but I'm looking for an obvious strategy to prove it. My try so far, I'm not quite sure if its correct, could you please check it?

If the function is continous at point $M_0$, we can say $\forall \epsilon>0$ $\exists \delta_0>0$ $\forall M \in D(f)$, the distance $\rho(M,M_0)<\delta_0$ in other words $M \in B_{\delta_0}(M_0) => |f(M)-f(M_0)|<\epsilon$

$0 < f(M_0)-\epsilon < f(M) < f(M_0)+\epsilon$

Here, I suppose, I need to give a proof by contradiction.

$\forall \delta>0$, no matter how small $\delta$ is, $\exists M' \in B_\delta(M_0) \cap D(f)$ s.t. $f(M')\leq 0$

let us fix $\delta=\frac{1}{n}$. $\exists N(\delta)\in \mathbf{N}$, that $\forall n>N$, $\exists \{ {M_n}^{'} \}_N^{\infty}{} \in D(f)$ sequence, s.t. $\rho(\{{M_n}^{'}\},M_0)<\delta=\frac{1}{n}$ and $f(\{{M_n}^{'}\})\leq 0$
because of continuity at $M_0$, we are getting $f(\{{M_n}^{'}\}) \rightarrow f(M_0)\leq 0$ which is a contradiction, because we have $f(M_0)>0$.QED.
Suppose $f(M_0) >0$, then there exists an $\epsilon >0$ such that $$f(M_0) \geq \epsilon.$$ Now let us work with $\epsilon/2$, from continuity, we know there exists a $\delta>0$ such that for each $M\in D(f) \cap B_\delta(M_0)$ we have $$|f(M_0) - f(M)|\leq \epsilon/2.$$ Now because $f(M_0) \geq \epsilon$, we get $$f(M) \geq \epsilon/2 >0.$$ The product of two positive numbers is positive, we have for each $M\in D(f) \cap B_\delta(M_0)$, $f(M_0)f(M) >0$.