Proving Log(x) is Continuous

  • Thread starter Yagoda
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  • #1

Homework Statement

Prove that [itex]f(x)=\log x[/itex] is continuous on [itex](0, \infty)[/itex] using that
(1) f is continuous at x=1 and
(2) [itex]\log(xy) = \log(x) + \log(y)[/itex]

Homework Equations

The definition of continuity: for all [itex]\epsilon >0[/itex], there exists a [itex]\delta>0[/itex] such that if [itex]|x-x_0| < \delta[/itex] then [itex]|f(x) - f(x_0)| < \epsilon[/itex].

The Attempt at a Solution

I think I've figured out how to do this using a more standard epsilon-delta proof, but it doesn't really make use of the two facts.
From what I can tell, it seems like you trying to be able to use the continuity at x=1 to "slide" the continuity down to 0 and up to infinity, but I'm not sure how to do this in a valid way. The only way I've managed to use fact 2 is rewrite things like [itex]\log x = \log(x \times 1) = \log(x)+\log(1) = \log(x)[/itex], which hasn't gotten me very far.
  • #2
Try replacing [itex]x_0[/itex] with xy for some y.
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  • #3
Look at ##f(x) - f(x_0)##, this becomes ##log(x) - log(x_0)##.
  • #4

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