We note that $H$ is in fact not Lipschitz continuous if this condition is violated.
a function continuous in space variables
More precisely, $f$ is just separately continuous.
The map $f$, which we know to be bounded, is also right-continuous.
a function continuous from the right
We follow Kato [3] in assuming $f$ to be upper semicontinuous.
Examples abound in which $P$ is discontinuous.
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