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I am trying to solve exercise 9.4.4 from Tao's Analysis I. It says:

Prove the following theorem:

Let $p$ be a real number. Then the function $f : (0, \infty) \to \mathbb{R}$ defined by $f(x) := x^p$ is continuous.

Hint: Prove that for all real numbers $p$ $\lim_{x \to 1} x^p = 1$. Then apply some basic facts about exponentiation.

The basic facts he refers to (he refers to the theorem by number) are:

Let $x,y>0, q,r \in \mathbb{R}$. Then:

  1. $x^q > 0$
  2. $x^{q+r} = x^q x^r$
  3. $(x^q)^r = x^{qr}$
  4. $x^{-q} = 1 / x^q$
  5. if $q>0$, then $x>y$ iff $x^q>y^q$
  6. If $x>1$, then $x^q>x^r$ iff $q>r$. If $x<1$, then $x^q>x^r$ iff $q<r$.

BTW the textbook defines exponentiation $x^\alpha$ for real $\alpha$ to be $\lim_{n \to \infty} x^{q_n}$, where $(q_n)_{n=1}^\infty$ is any sequence of rational numbers converging to $\alpha$.

I think I can prove it using other means - first proving that for any integer $n$ $f(x) = x^n$ is continuous, then prove that for any positive integer $n$ $f(x) = x^{1/n}$ is continuous, then prove that for any rational $p$ $f(x) = x^p$ is continuous, then prove continuity for real $p$.

But I have no idea how to do it using the author's hint. I have proved that $\lim_{x \to 1} x^p = 1$, but I've got no idea what to do with this fact. Please help me figure it out.

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You want to show that, for every $a>0$, $$ \lim_{x\to a} f(x) = f(a) $$ i.e., $$ \lim_{x\to a} x^p = a^p $$ and you want to infer this from the case $a=1$. Observing that ${x^p}/{a^p} = (x/a)^p$, can you prove that $$ \lim_{x\to a} \frac{x^p}{a^p} = 1 $$ and conclude from there?

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  • $\begingroup$ This helped me, I have not solved it, and the steps you suggested turned out to be very simple. I wonder why I missed this and what general technique of proving things I should learn to see such proofs. Thank you. $\endgroup$ – CrabMan Sep 27 '18 at 10:15
  • $\begingroup$ @CrabMan I am not aware of a general technique, but in general trying to reverse-engineer the problem helps (to start a proof; the proof itself should not go backwards). So, on rough paper, start with the conclusion, and try and poke/massage (no need to be too rigorous) it until you get something that resembles your assumptions. Then, once you have an idea of a path to go the other way around, you can formalize the whole thing and fill the gaps. $\endgroup$ – Clement C. Sep 27 '18 at 19:19
  • $\begingroup$ Btw I've asked Terry Tao by email, he replied that indeed this property is supposed to be included in the list, and it is missing by mistake. $\endgroup$ – CrabMan Sep 28 '18 at 6:16

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