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## Homework Statement

Prove that [itex]f\left(x\right)=\log_{a}x[/itex] is continuous for all [itex]\mathbb{R}[/itex].

## Homework Equations

[/B]I must find a [itex]\delta>0\in\mathbb{R}[/itex] for a given [itex]\varepsilon>0[/itex]

such that

[tex]

\left|x-x_{0}\right|<\delta\Rightarrow\left|\log_{a}x-\log_{a}x_{0}\right|<\varepsilon.

[/tex]

## The Attempt at a Solution

.[/B]I tried to use a direct proof solving [itex]\left|\log_{a}x-\log_{a}x_{0}\right|<\varepsilon[/itex] for [itex]x[/itex]. But this gives rise to a pair of values: [itex]\delta=x_{0}\left(a^{\varepsilon}-1\right)[/itex] and [itex]\delta=x_{0}\left(a^{-\varepsilon}-1\right)[/itex]. When I use them to build [itex]\left|x-x_{0}\right| < \delta[/itex] from the inequality

[tex]

-\delta<x-x_{0}<\delta

[/tex]

I see myself in big trouble, as there is no way to generate [itex]\left|\log_{a}x-\log_{a}x_{0}\right| < \varepsilon[/itex].

Can someone give me a hint, or a new strategy? I have searched for help in other forums, but hints are very sophisticated to follow.

Thanks in advance.