Useful examples of pathological functions

The Weierstrass function is continuous everywhere and differentiable nowhere.
The Dirichlet function (the indicator function for the rationals) is continuous nowhere.
A modification of the Dirichlet function is continuous at all irrational values and discontinuous at rational values.
The Devil's Staircase is uniformly continuous but not absolutely. It increases from 0 to 1, but the derivative is 0 almost everywhere.

Also: Conway base 13 function.

This function has the following properties.
1. On every closed interval $[a, b]$, it takes every real value.
2. It is continuous nowhere.

Here's an example of a strictly increasing function on ℝ which is continuous exactly at the irrationals.

Pick your favorite absolutely convergent series ∑a_n in which all the terms are positive (mine is ∑1/2ⁿ) and your favorite enumeration of the positive rationals: ℚ={q₁,q₂,...}. For a real number x, define f(x) to be the sum of all the a_n for which q_n ≤ x.

Have also a look here:
https://mathoverflow.net/questions/22189/what-is-your-favorite-strange-function

The Dirac delta "function." It's not a "function," strictly speaking, but rather a very simple example of a distribution that isn't a function.