6. Limits, Continuity, and Differentiation
6.6. A Function Primer
Here we want to list some functions that illustrate more or less
subtle points for continuous and differentiable functions. These
functions are all difficult, in one sense or another, but should
definitely be part of the repertoire of any math student with
an interest in analysis.
Examples of Continuous and Differentiable Functions

Dirichlet function: A function that is not continuous at any point in R 

Countable discontinuities: A function that is continuous at the irrational numbers and discontinuous
at the rational numbers. 

C^{1} function: A function that is differentiable, but the derivative is not continuous. 

C^{n} function: A function that is ntimes differentiable, but not (n+1)times differentiable 

C^{inf} function: A function that is not zero, infinitely often differentiable, but the nth
derivative at zero is always zero. 

Weierstrass function: A function that is continuous everywhere and nowhere differentiable in R. 

Cantor function: A continuous, nonconstant, differentiable function whose derivative is zero
everywhere except on a set of length zero. 