the space of all continuous functions on X
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.
To be precise, A is only left-continuous at 0.
a function continuous from the right
We follow Kato  in assuming f to be upper semicontinuous.
Examples abound in which P is discontinuous.