Number of discontuinity Two to one functions from $\mathbb{R}$ to $\mathbb{R}$

Question

Let's call a function $ f: \mathbb{R} \rightarrow \mathbb{R} $ Two to one if for any $y \in Im(f)$ there is exactly two distinct $x_1,x_2 \in \mathbb{R}$ such that $f(x_1)=f(x_2)=y$

Is it true that every Two to one function with domain $\mathbb{R}$ must have infinite number of discontinuity?

P.S: As answered before here it must be discontinous but can it have finite number?