my experience with world-renowned experts in pure mathematics is that they are much more careful than the average bear to explicitly qualify inaccurate things as inaccurate, because their discipline requires them to be very clear about precisely what they are saying
discontinuity of a function at x does not, according to the usual definition of 'continuity', require the function to have a value at x; indeed, functions that fail to have a value at x are necessarily discontinuous there, precisely because (as you say) they are not continuous there. https://en.wikipedia.org/wiki/Continuous_function#Definition...
there are other definitions of 'discontinuous' in use, but i can't think of one that would give the result you claim
discontinuity of a function at x does not, according to the usual definition of 'continuity', require the function to have a value at x; indeed, functions that fail to have a value at x are necessarily discontinuous there, precisely because (as you say) they are not continuous there. https://en.wikipedia.org/wiki/Continuous_function#Definition...
there are other definitions of 'discontinuous' in use, but i can't think of one that would give the result you claim