Consider an alternate definition of continuity

[lowasser]

Classical analysis defines the word "continuity" by itself as pointwise continuity.

Bishop, and some other constructive analysis texts I've read, use a different definition of "continuous function" that is classically equivalent, which I'll call local uniform continuity: that a function is uniformly continuous on every closed interval (in the reals), or (in a more general metric space) on every totally bounded subset.

I think we should use this classically equivalent but much more constructively useful definition for the word "continuous," and reserve "pointwise continuous" for the direct translation of the classical definition.

This is at least partially me going, "I absolutely do not want to write locally-uniformly-continuous in practically every function name."

[fredrik-bakke]

This seems like an entirely reasonable suggestion to me 👍 Pointwise continuity is after all well known under the name "pointwise continuity".