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===A function that does not satisfy
[[File:Kakutani non.svg|thumb|150px|A function without fixed points]]The requirement that ''φ''(''x'') be convex for all ''x'' is essential for the theorem to hold.
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=== A function that does not satisfy close graph ===
Consider the following function defined on [0,1]:
:<math>
\varphi(x)=
\begin{cases}
3/4 & 0 \le x < 0.5 \\
1/4 & 0.5 \le x \le 1
\end{cases}
</math>
The function has no fixed point. Though it satisfies all other requirements of Kakutani's theorem, its graph is not closed; for example, consider the sequences ''x<sub>n</sub>'' = 0.5 - 1/''n'', ''y<sub>n</sub>'' = 3/4.
==Alternative statement==
Some sources, including Kakutani's original paper, use the concept of [[Hemicontinuity#Upper hemicontinuity|upper hemicontinuity]] while stating the theorem:
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