Are Continuous Functions Always Differentiable?

Question

Are continuous functions always differentiable? Are there any examples in dimension $n > 1$?

Answer 1

No. Weierstraß gave in 1872 the first published example of a continuous function that's nowhere differentiable.

Answer 2

For a nice simple example of an everywhere continuous, nowhere differentiable function it's hard to beat this example of John McCarthy.

Answer 3

No, consider the example of $f(x) = |x|$. This function is continuous but not differentiable at $x = 0$.

There are even more bizare functions that are not differentiable everywhere, yet still continuous. This class of functions lead to the development of the study of fractals.

Answer 4

The Wiener process is a continuous everwhere, but differentiable nowhere function (quite an impressive beast by the way...)

Answer 5

An interesting fact is that most (i.e. a co-meager set of) continuous functions are nowhere differentiable. The proof is a consequence of the Baire Category theorem and can be found (as an exercise) in Kechris' Classical Descriptive Set Theory or Royden's Real Analysis.