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Let's assume we are given an arbitrary function $f : \mathbb{R} \to \mathbb{R}$, and for the purposes of this question, let's assume we know nothing about the differentiability of $f$, i.e. we have no pre-requesite knowledge about the differentiability of $f$

Now for a function to be differentiable at some point $a$, the following limit must exist and be finite

$$f'(a) = \lim_{h \to 0} \frac{f(a+h)-f(a)}{h}$$

Furthermore a function is not differentiable at some point $a$, if :

  1. $f$ is discontinuous at $a$, i.e $\lim_{x \to a}f(x) \neq f(a)$
  2. $f'(a)$ does not exist
  3. $f$ has a vertical tangent at $a$, i.e. $\lim_{h\to 0} \frac{f(a+h)-f(a)}{h} = \infty$

If we do not know anything about the differentiability of $f$ beforehand, how do we go about finding the set of values (which could be zero, finite or infinite) for which $f$ is not differentiable?

$S = \{a \ | a \in \mathbb{R} \text{ and} \ a \text{ satisfies conditions 1 or 2 or 3 above}\}$

What I'm trying to ask in a nutshell:

In Real Analysis, what methods/techniques are commonly used to prove an arbitrary function $f$ is differentiable $\forall \ a \in \mathbb{R}$, or to find the set of values $S$, where $f$ is not differentiable.?

I say specifically an arbitrary function $f$, as there might be different techniques used for algebraic functions as opposed to say transcendental functions. From what I've encountered in introductory Real Analysis, this is not something that is gone into great depth or detail.

What I want to know is how did Mathematicians prove that functions such as $f(x) = e^x$ are differentiable $\forall x \in \mathbb{R}$, and how did they prove that functions such as $f(x) = \sqrt[3]{x}$, are not differentiable at certain values of $x$ on the Real Field. (I've given a simple example for the second case, however I'm sure that there are functions, which are not differentiable at many points in $\mathbb{R}$).

Elementary techniques such as Mathematical Induction fall apart when working with $\mathbb{R}$. Furthermore for finding the set of values $S$ where a function is not differentiable, I'm sure they didn't just find them heuristically by testing values of possible interest for a given function. So what techniques are commonly used to formulate these proofs?

Perturbative
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2 Answers2

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This is a pretty big question but I can do my best.

First, you're right, it does depend on the function. As for a general function $f(x)$, the only general method to show it is differentiable is to use the definition.

BUT, there are a few short-cuts. First, we can prove that all polynomials are differentiable. We can also get a "library" of differentiable functions, such as $\sin x$ or $e^x$. Finally, we can prove facts such as: if $f$ and $g$ are differentiable, then $f+g$ is differentiable, and $fg$ is differentiable, and $f/g$ is differentiable as long as $g\neq 0$.

Some functions' differentiability are easier to prove than others. Take $f(x)=e^x$. By writing $$\left| \frac{f(x+h)-f(x)}{h}\right|=\left| \frac{e^{x}e^h-e^{x}}{h}\right|=e^{x}\left|\frac{e^h-1}{h} \right|$$ we see that the differentiability of $e^x$ comes down to the fact that $$\lim_{h\to 0}\left|\frac{e^h-1}{h}\right|=1.$$ This is pretty tedious to do (remember we cannot use L'Hospitals as that involves taking the derivative of $e^x$ in the first place). A full proof is here.

As for knowing how to find the points of discontinuity, you often can make a educated guess, and then rigorously prove your claim. It is when you get into functions without graphs that it gets harder to make a conjecture, but even then there may be an obvious choice.

Hope that helps somewhat.

Community
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pancini
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A truly "arbitrary" function can't even be specified with finitely many symbols, and there is no way to prove whether it is differentiable at a point. But most of the functions that we usually deal with are given by formulas, i.e. built up from simple building-blocks such as constants, the variable $x$, and some special functions such as $\sin$ and $\exp$, using the operations of arithmetic and the composition of functions. We can then use the theorems that say, e.g., that the sum of differentiable functions is differentiable. Given such a formula, you can usually tell what will be the problematic points where these theorems will not apply, and then look at those in more detail.

For example, if you have a formula involving a square root, there will be a problem if the expression inside the square root is negative (then the function is not defined unless you allow complex numbers), and there may be a problem at a point where the expression inside the square root is $0$ (even though it may be positive on both sides of that point).

Robert Israel
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