# Sufficient condition for a continuous function to be differentiable

Let $$U$$ be an open set of $$\mathbb R$$, and suppose that $$f:U\to\mathbb R$$ is continuous. It is well-known that $$f$$ need not be differentiable at every point in U; in fact, the Weierstrass function gives us an example of a function that is continuous everywhere but differentiable nowhere. However, continuous functions are still much better behaved than "general" functions, and so it seems plausible that with some additional hypotheses on $$f$$, we can guarantee that $$f$$ is differentiable everywhere (or, at least, differentiable almost everywhere).

With this in mind, I ask the following question:

What examples are there of theorems which give sufficient conditions for $$f$$ to be differentiable (or differentiable almost everywhere)?

The fact that the Weierstrass function has a special name seems to indicate that not being differentiable is the exception, and your question indicates a similar perception. This is, however, not correct, in some quite precise sense:

In the book 'Real and Abstract Analysis' (see (**) below) you will find the following result (Theorem 17.8 in my edition):

In the real Banach Space $$C = C ([0,1])$$ (*) let $$D= \{f\in C: \text{left and right hand side derivative of f are both finite for} some \, x\in [0,1]\}$$ Then $$D$$ is of the first category in the complete metric space $$C$$, so the set of all continuous functions on $$[0,1]$$ which have at least one infinite right derivative at every point of $$[0,1]$$ is dense in $$C$$.

To hint at some answers to your question, despite of this result: monotonic, concave (or convex), absolutely continuous (AC) functions, Lipschitz continuous functions and functions of bounded variation are typical classes of functions which have derivatives in 'many' points. (only concave, Lipschitz and AC functions being a subset of $$C$$, though (***)). - I leave it to you though to explore the pertinent results.

(*) in that book you will find a notation like $$C^r$$, which I found confusing at first - it only indicates, though, that they are looking at real valued functions, with $$C$$ being reserved for complex valued ones.

(**) Hewitt, Edwin; Stromberg, Karl, Real and Abstract Analysis. A modern treatment of the theory of functions of a real variable, Berlin-Heidelberg-New York: Springer-Verlag. VIII, 476 p. with 8 fig. (1965). ZBL0137.03202.

(***) I'm a bit sloppy here. A concave function on $$[a,b]$$ will be continuous on $$(a,b)$$ (and locally Lipschitz continuous: Every convex function is locally Lipschitz ($\mathbb{R^n}$))