>> 1. Wrap char-displayable-p in (with-selected-frame ...).
>> 
>> 2. Give char-displayable-p a frame argument, and wrap its body in
>>    (with-selected-frame ...).
>> 
>> 3. Drop the frame argument altogether (leaving (with-selected-frame
>>    ...) to callers of these functions).
>
> I prefer (1), since we cannot change char-displayable-p itself (it's
> used in too many places, including during startup).
>
>> Option (3) seemed simplest to me, and also at the least risk of
>> introducing confusion concerning the name of the new function: at a
>> glance, "char-displayable-on-frame-p" might suggest incorrectly that it
>> does the analogue for the given frame of what char-displayable-p does
>> for the selected frame.
>
> I acknowledge the potential confusion, but I think we can countermand
> that in the doc string (without even mentioning char-displayable-p).

Thanks, I've revised to (1).  Please see attached.