For $X\sim\mathcal{N}(0,1)$, find measurable functions $g,h$ s.t. $g(X),h(X)$ are uncorrelated 
Let $X\sim\mathcal{N}(0,1)$ a random variable. Find measurable function $g,h:\mathbb{R}\to\mathbb{R}$ such that $g(X),h(X)$ are uncorrelated.

I would like to confirm my answer:
Let $g,h:\mathbb{R}\to\mathbb{R}$ defined by $g(t)=1,h(t)=t$. $g$ and $h$ are measurable because ther're continuous.
$$
\\ \mathbb{E}(g(X)h(X))=\mathbb{E}(h(X))=\mathbb{E}(h(x))\cdot1=\mathbb{E}(h(X))\cdot\mathbb{E}(g(X)) \ 
$$

Your answer is correct, but rather trivial.  My guess is that whoever posed the question was hoping for a more interesting example than that one.
Another one to think about might be $g(x) = |x|$, $h(x) = \operatorname{sign} x$.  And you might like to prove that they are not only uncorrelated but actually independent.
