2

Definition: A function $f:\mathbb{R}\to \mathbb{R}$ is called faintly continuous in $x$ if there are two series $x_n < x < y_n$ with $\lim_{x_n \to x} f(x_n) = \lim_{y_n \to x} f(y_n) = f(x)$.

Conjecture: Any funtion $f:\mathbb{R}\to \mathbb{R}$ is faintly continuous everywhere except at an at most countable set of discontinuities.

Is that correct? What other statements can be made related to the definition? Is there research on this and similar extremely weak properties of real (or complex) functions? Oh, and if the above has a different English name already, I'd also care to hear!

Math ends here, background for those who care: The direct translation from the German article this conjecture is from (it actually was a theorem there, but the proof was ... strange) would be "weakly" continuous, but that term is already used otherwise. I always thought the conjecture was wrong. I can't offer references since this really is just a nostalgic flashback to some book I read 15+ years ago and a problem I never really resolved.

Desiato
  • 1,610

1 Answers1

1

(Two comments now posted as an answer, since no one else has said anything.)

Results such as this have been sharpened and generalized nearly to the extent that the Pythagorean Theorem has been. For early work, see my answer to the math overflow question A search for theorems which appear to have very few, if any hypotheses.

For later work and generalizations, google things like cluster set, radial limit, curvilinear cluster set, non-tangential boundary values, tangential boundary values, tangential cluster set, Stolz extensions, boundary functions, directional cluster sets, tangential boundary behavior, rectifiably ambiguous points, etc. In fact, the Unabomber's mathematical research was in this area.

Community
  • 1
Dave L. Renfro
  • 36,843