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Non-continuous Functions

A function can still be continuous and not be differentable at a point. For example, the absolute value function |x| is continuous at x = 0, but the deriviatve is not defined at that point. Here we give an example of a function, the Weierstrass function W, which is continuous everywhere but differentiable nowhere.

A function need not be continuous anywhere, for example, the function:

where Q is the set of rationals.

It can also get worse: Thomae's function χ(x) is a function which is continuous at all irrationals and discontinuous at all rationals:

Figure 1 shows a plot of the function χ(t) while Figure 2 shows a zoom on the black point (1/2½, χ(1/2½)) = (1/2½, 0) (because the square root of two is irrational). If the points in Figure 1 are too fine, you may also look at this view.

Figure 1. Thomae's function.

Figure 2. A zoom on Thomae's function at 1/2½.