In functional analysis, a branch of mathematics, Michael selection theorem is a selection theorem named after Ernest Michael. In its most popular form, it states the following:

Michael Selection Theorem—Let X be a paracompact space and Y be a separable Banach space. Let ${\displaystyle F\colon X\to Y}$ be a lower hemicontinuous set-valued function with nonempty convex closed values. Then there exists a continuous selection ${\displaystyle f\colon X\to Y}$ of F.

Conversely, if any lower semicontinuous multimap from topological space X to a Banach space, with nonempty convex closed values, admits a continuous selection, then X is paracompact. This provides another characterization for paracompactness.

The function: ${\displaystyle F(x)=[1-x/2,~1-x/4]}$, shown by the grey area in the figure at the right, is a set-valued function from the real interval [0,1] to itself. It satisfies all Michael's conditions, and indeed it has a continuous selection, for example: ${\displaystyle f(x)=1-x/2}$ or ${\displaystyle f(x)=1-3x/8}$.

The function

${\displaystyle F(x)={\begin{cases}3/4&0\leq x<0.5\\\left[0,1\right]&x=0.5\\1/4&0.5<x\leq 1\end{cases}}}$

is a set-valued function from the real interval [0,1] to itself. It has nonempty convex closed values. However, it is not lower hemicontinuous at 0.5. Indeed, Michael's theorem does not apply and the function does not have a continuous selection: any selection at 0.5 is necessarily discontinuous.

Michael selection theorem can be applied to show that the differential inclusion