Function defined on R and discontinuous at all points Apart from the Dirchet function ( equals 1 for rational x and 0 for irrational x), and similar constructs (equals some constant a for rational x and b for irrational , where a!=b) , what are the other functions which are defined on all real numbers and discontinuous at all points?
Are there infinite such functions? 
 A: Let $f(x)$ be a continuous function defined on $\mathbb{R}$, and let $c\neq0$ be a real number.  The function
$$g(x):=\begin{cases}f(x)&\text{ if }x\in\mathbb{Q} \\f(x)+c&\text{ otherwise}\end{cases}$$
is discontinuous for all $x\in\mathbb{R}$.
A: There are as many such functions as all functions from $\Bbb R \to \Bbb R$, which is $2^{\mathfrak c}$  We can define a set of functions $$f(x)=\begin {cases}0& x \text { rational} \\ 1& x \text { algebraic but not rational}  \\ \text {something}&x \text { transcendental} \end {cases}$$
As there are uncountably many transcendental numbers, there are $2^{\mathfrak c}$ functions in this set.  As the rationals and irrational algebraics are dense, this is nowhere continuous.  The subtext is that every nice property of functions is rare.   I suspect one could define a measure on the space of functions and show that those that are continuous somewhere have measure zero. Certainly those that are continuous everywhere are of measure zero as there are only $\mathfrak c$ of them.
