Prove that $g(x)=e^{-x^2}$ has a unique fixed point on the interval [0,1] Hello I need help with this question, 
Prove that $g(x)=e^{-x^2}$ has a unique fixed point on the interval [0,1]

The continuous function $f(x) = e^{-x^2} - x$ is positive at 0 and negative at 1, so by the Intermediate Value Theorem there is a point $a \in [0,1]$ with $e^{-a^2} - a = 0$ or $e^{-a^2} = a$. This fixed point is unique because $e^{-x^2}$ is monotonically decreasing on $[0,1]$.
