Discussion Overview
The discussion revolves around the concept of continuity to the right for the repartition function F, specifically examining whether the convergence of F(x + 1/n) to F(x) is sufficient to establish continuity for any sequence converging to x from the right. The scope includes theoretical aspects of continuity in probability functions.
Discussion Character
Main Points Raised
- One participant questions whether proving F(x + 1/n) converges to F(x) is sufficient for continuity to the right for any sequence converging to x, suggesting that continuity to the right should apply to any sequence {a_n} in (x, +∞).
- Another participant agrees that if the proof relies solely on the convergence of the sequence to zero, then it should not depend on the specific choice of 1/n as an example.
- A later reply indicates that the increasing nature of the function F allows the proof to extend beyond the 1/n sequence, implying that the properties of increasing functions support the argument.
- Further clarification is provided that since F is increasing, values of F for points between 1/n and 1/(n+1) must also fall between F(1/n) and F(1/(n+1), suggesting that the epsilon/delta argument for the 1/n sequence applies to other sequences converging to zero.
Areas of Agreement / Disagreement
Participants express differing views on whether the convergence of F(x + 1/n) to F(x) is sufficient for continuity to the right for any sequence. While some acknowledge the increasing nature of F as a factor that may support the argument, there is no consensus on the sufficiency of the proof for all sequences.
Contextual Notes
The discussion highlights the potential limitations of the proof's reliance on the specific sequence 1/n and the implications of the increasing property of the function F. There is an unresolved consideration regarding whether the argument holds for all sequences converging to x.