Surjectiveness of standard-normal c.d.f.

Question

Let $\phi:\mathbb R \to (0,1)$ be a function defined as $\phi(y)=\int_{-\infty}^y\dfrac{1}{\sqrt{2\pi}}e^{-\dfrac {x^2}{2}}dx , \forall y\in \mathbb R$ , then is it true that $\phi$ is surjective ? If yes, then what is the proof ?

Answer 1

We have $\lim_{y\to-\infty}\phi(y)=0$ and $\lim_{y\to\infty}\phi(y)=1$. Moreover, the function $\phi$ is continuous. Thus, by the Intermediate Value Theorem, the function $\phi$ is surjective.