I'm looking to find out whether or not there exists a continuous function $f \colon \mathbb{R}\to\mathbb{R}$ such that for $\textbf{any}$ $\alpha\in\mathbb{R}$ the equation \begin{equation*} f(x)=\alpha \end{equation*} has an uncountable number of solutions.
I am tempted to believe that such a function doesn't exist, but I can't prove it. Despite this, I was able to find functions satisfying weaker conditions. For example for the function $f \colon \mathbb{R}\to\mathbb{R}$, $f(x)=x\sin x$ and for any $\alpha$ the equation \begin{equation*} f(x)=\alpha \end{equation*} has a countable number of solutions.
An idea to build the desired function would be to try to replicate the behaviour of $\sin\frac{1}{x}$ around $0$, especially at plus minus infinity, but I can't quite get my head around this.
Consider $\pi_1 \circ f$, where $f : [0, 1] \to [0, 1] \times [0, 1]$ is a space-filling curve and $\pi_1 : \mathbb{R}^2 \to \mathbb{R}$ is projection onto the first coordinate, i.e. $\pi_1(x, y) = x$. It doesn't give you everything (since it only has the range $[0, 1]$), but it's a start.