In the early 18th and 19th centuries, most mathematicians thought that continuous functions were differentiable everywhere except perhaps at a few various points. In 1806, Andre Marie Ampere tried, and failed, to prove this result, which was subsequently known as Ampere’s Theorem. He also failed to provide a counterexample. Lagrange later attempted to prove the result to no avail.
In 1872, Weierstrass shocked the world by exhibiting a class of functions that were continuous everywhere, but differentiable nowhere. Since then, many other examples have been discovered. In fact, Weierstrass wasn’t the first to discover such a function. Bolzano constructed one geometrically as early as 1830. Indeed, Bolzano was far ahead of his contemporaries in almost all of mathematical thought. Unfortunately for him, however, his work was not circulated and by the time it was discovered other work had surpassed it.
Since there are now many examples of such pathological functions, how many are there exactly? One way to answer this question is to apply Baire’s category theorem. Baire split sets into one of two categories. Sets of the first category (also called meager in modern terminology) are considered small or thin. All other sets are sets of the second category. Using Baire’s category theorem (which says that a complete metric space is of the second category), we can show that the set of continuous functions which have a derivative at at least one point is meager.
Hence the set of continuous nowhere differentiable functions (being the complement of that set) is not meager! That is, this set is huge. So if someone hands you a random continuous function, the odds are that it is nowhere differentiable. Hence these monstrosities are the norm, not the exception.
You can read more about this topic in this excellent master’s thesis.
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