# Cardinality of all inverse functions (bijections) defined on: $\mathbb{R}\rightarrow \mathbb{R}$

What is the cardinality of all inverse functions defined on: $\mathbb{R}\rightarrow \mathbb{R}$?

easy to calculate the upper bound which is $2^\aleph$. ($\aleph$ is the cardinality of the continuum)

I need to find a lower bound. Probably by finding an injective function and deduce by Cantor–Bernstein–Schroeder theorem that the cardinality is $2^\aleph$.

Hint: There are $\aleph$ positive reals. For any subset $A$ of the positive reals, define a bijection $\phi_A$ from $\mathbb{R}$ to $\mathbb{R}$ as follows. Let $\phi_A(0)=0$. If $a\in A$, let $\phi_A(a)=a$ and $\phi_A(-a)=-a$. If $a\not\in A$, let $\phi_A(a)=-a$ and $\phi_A(-a)=a$.