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It seems to me that the definition of a critical point is a bit of a polarizing subject. Some sources say that a function $f(x)$ has a critical point at $x=p$ if and only if $f'(p)=0$, while others say that $f'(p)$ must be either $0$ or undefined. My question is, of what significance is it to any field of mathematics whether we say that a nowhere differentiable but everywhere continuous function (such as the Weierstrass function) has critical points nowhere or everywhere? It seems like a pretty extreme decision but I'm curious to know if it's all just for argument's sake or if it actually effects important calculations.

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Colloquially speaking, I think of a critical point to much more commonly mean $f'(p)=0$, and then look at the behavior of the function nearby to assess it. Technically, a constant function satisfies $f'(p)=0$ in intervals, but I don't think of these as critical points. Much less often, I think of things like vertical asymptotes and essential singularities as being critical points. Although these behaviors are important, they are usually emphasized for what they are rather than the generic "critical point." I definitely don't think of, say, Brownian motion as having critical points everywhere.

If a function is continuous, but nowhere differentiable in a neighborhood, this is always stated explicitly. If you were to say continuous and every point is a critical point, people would probably just think that was a weird way of describing a constant function.

From a research perspective, I can't see any issues with the language used how it is already.

If I had to guess, I think the reason critical point is played up in early calculus courses is because students don't have the maturity to understand what's going on around blow up points. It's common for a student to set up $f'(x)=0$, set the numerator equal to zero, and ignore points where the denominator blows up. They then will say they have found the absolute max of the function because they "looked at all critical points." A mature mathematician wouldn't make this mistake so prefers more specific terminology to describe blow ups.

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  • $\begingroup$ Thank you for taking the time to answer so thoroughly. I have a couple of questions though, what do you mean by blow-up points and how can a denominator blow up? This is new terminology to me. $\endgroup$ Apr 7, 2020 at 14:32
  • $\begingroup$ Sorry, I was tired when I wrote that. By "denominator blowing up" I meant the function blowing up because the denominator goes to zero. $\endgroup$
    – ProfOak
    Apr 7, 2020 at 15:15
  • $\begingroup$ I thought so. Also, when you say, "A mature mathematician wouldn't make this mistake so prefers more specific terminology to describe blow ups." Where can I read about the terminology a mature mathematician would use? $\endgroup$ Apr 7, 2020 at 15:17

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