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I was watching Lecture 6 of 18.01 (Single Variable Calculus) on ocw.mit.edu.

This statement came from the lecture and I can't make sense of it. Could anyone help? Thanks!

Statement: a^x is defined for all x by "filling in" by continuity

bryanlovesmechanics
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    It's hard to give a good answer at this point. The issue is that you want to say that, for example, the square root of $a$ exists. How do you know it exists? For example, how do you know there actually is a number multiplied by itself that equals 2? I'm assuming they want to use a continuity argument (probably by way of intermediate value theorem) to argue this. The issue is how do you show $a^x$ is continuous if you don't even a priori know it's well defined? So you'd need to define the exponential function, and the logarithm, and proceed. It's probably best to just ignore it for now... – Andrew Dec 10 '22 at 07:42
  • We can define $a^x$ for integers and rational numbers. For example, $a^3=aaa$, $a^{-2}=\frac{1}{aa}$ and $a^{\frac{3}{2}}=\sqrt{aa*a}$. But $a^{\pi}$ doesn't have an intuitive definition so we define it by considering that we can find rational numbers arbitrarily close to $\pi$, and continuity of $a^x$ allows us to say that $a^{\pi}$ exists. – John Douma Dec 10 '22 at 09:01
  • @JohnDouma I disagree. If you are asking about whether it exists for irrational numbers, how can you say it's continuous? And there's no a priori reason to think that rational powers exist either. To proceed, you need to define the exponential and logarithm, and prove continuity of them. But then there would be no question even about whether $x\mapsto a^x$ is well defined! – Andrew Dec 10 '22 at 09:21
  • @AndrewZhang I haven't seen the lecture being discussed, but we usually state without proof, that the exponential functions are continuous in beginning courses. The point of my comment wasn't to justify the continuity of the exponential function but to point out that we have clear intuitive definitions of integer and rational exponents. I suspected that the video meant what I meant when it said "filling in by continuity". – John Douma Dec 10 '22 at 09:41

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