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## Homework Statement

Let [tex] \displaystyle f:{{\mathbb{R}}^{n}}\to \mathbb{R}[/tex] a continuous function. Proove that:

If [tex] \displaystyle f\left( p \right)>0[/tex] then there's a ball [tex] \displaystyle {{B}_{p}}[/tex] centered at p such that [tex] \displaystyle \forall x\in {{B}_{p}}[/tex] we have [tex] \displaystyle f\left( x \right)>0[/tex].

## Homework Equations

## The Attempt at a Solution

f is continuous, that is:

[tex] \displaystyle \forall \varepsilon >0[/tex] [tex] \displaystyle \exists \delta >0[/tex] such that [tex] \displaystyle f\left( B\left( p,\delta \right) \right)\subset B\left( f\left( p \right),\varepsilon \right)[/tex].

Let f(p)>0, then I cannot conclude anthing with the above hypothesis. What's missing?

Thank you!