TITLE: Cardinality of set of all everywhere-discontinuous functions
QUESTION [12 upvotes]: The question is to find the cardinality of the set of all everywhere-discontinous real-valued functions of real variable.
My intuition tells me there are $2^c$ such functions, but I can't seem to find an injection from the set of all functions to the set of everywhere-discontinuous functions.
Any help would be appreciated.
$c$ here denotes the cardinality of continuum (for an example, the cardinality of set of all real numbers).

REPLY [6 votes]: Here is a completely constructive mapping that is easily proven to have the desired property.
For each function $f : \mathbb{R} \to \mathbb{R}$, let $g(x) = \tanh(f(x)) + \cases{1 & if $x\in\mathbb{Q}$ \\ -1 & otherwise}$ for each $x \in \mathbb{R}$. Then $g : \mathbb{R} \to \mathbb{R}$ and $g$ is everywhere discontinuous.