Let $f(x)$ be a continuous real function s.t $f(x_0) > 0$
Prove: There is some interval of the form $(x_0 -\delta, x_0 + \delta)$ where $f$ is positive.
Proof:
Since $f$ is continuous: $\forall \,{\epsilon > 0}\,\, \exists \,{\delta>0}$ s.t. $|x- x_0|<\delta \implies |f(x) - f(x_0)| < \epsilon$
By contradiction suppose there is no interval $(x_0 - \delta, x_0 + \delta)$ where $f(x)$ is positive. This means that $f(x_0) - \epsilon < f(x) < f(x_0) + \epsilon < 0$. Hence we have a contradiction since $\epsilon$ and $f(x_0)$ are both greater than zero.
- Is this correct?
- Could someone provide a non-contradiction proof?