Let $X$ be a standard normal random variable and $c$ be a real number. Show that $$\mathbb E((X-c)_+) = g(c) := \frac{1}{\sqrt{2\pi}} e^{-c^2/2}-cN(-c).$$
I am curious though because I thought if $c = 0$, it implies that expectation should be $$ \Bbb E(X_+) = \frac{\sqrt2}{\sqrt\pi} $$
And that doesn't seem to be the result.