Okay, so in class today we went over proving continuity by the formal definition of limits, that is

$$\forall \epsilon \gt 0, \exists \delta \gt 0 \mid \lvert x-a \rvert \lt \delta \implies \lvert f(x)-f(a)\rvert\lt\epsilon$$

So, that's all fine. But my professor gave an example where $f(x)=\ln(x)$ and he tried to show continuity at $x=2$.

Now, he proved it using the MVT:

$$\lvert\ln(x)-\ln(2)\rvert = \lvert f'(c) \left(x-2\right)\rvert$$

Am I crazy to think this cannot be used? For the MVT, you are assuming continuity, but we are trying to prove continuity!

Also, if MVT cannot be used, can someone show how to prove this?

  • 2
    $\begingroup$ MVT shouldn't be used as 2 would be part of the interval where continuity is part of the basis of being able to use the theorem. $\endgroup$ – JB King Sep 4 '13 at 17:42
  • $\begingroup$ The Mean value theorem asserts continuity as part of the hypothesis, so we shouldn't use it to try and prove continuity! $\endgroup$ – Euler....IS_ALIVE Sep 4 '13 at 17:50
  • $\begingroup$ I think we could use $|\log(x) - \log2| = |\log(x/2)|$. This should help choose our $\delta$ $\endgroup$ – Hawk Sep 4 '13 at 18:04
  • $\begingroup$ I assume you defined logarithm using integrals. Now, just quote fundamental theorem of calculus. $\endgroup$ – Moishe Kohan Sep 4 '13 at 18:16

Your right that it's sort of circular to assume differentiability. But telling you how to do it is a bit harder unless we know what you think of as a natural log.

One of the ways I like seeing the natural log defined as a function is $\ln x = \displaystyle \int_1^x \frac{1}{t} \mathrm{d}t$, in which case it is trivially differentiable by the fundamental theorem of calculus, and therefore continuous. If this is what your professor/teacher has done, then it's logically sound, but a terrible example.

Other common ways of defining the natural log is $\ln x = \displaystyle \lim_{k \to 0} \dfrac{x^k - 1}{k}$, the inverse of $e^x$, as a power series, etc. And each of these has their own way of determining continuity. But a very common theme is to first show that the exponential is continuous, and then get the corresponding result. Interestingly, this also often relies on differentiation instead of direct computation.


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