Limits and continuous 2

yma16 · 发表于 2006-10-30 19:05:05

Prove of disprove that if a function has one side limit at every real numbers, the function is almost continuous everywhere.

HF: · 发表于 2006-10-30 19:23:09

By 'one side limit', do you mean that 'both left side limit and right side limit?' or 'either left side limit or right side limit'?

If it is the former, it is not too hard to show that the the function is continuous everywhere except at countable many points.

The conclusion will still be true if we relax the condition to 'left side limit exists at every point' or 'right side limit exists at every point'.

It is not clear to me whether it is true if the condition is even weaker, that 'either left side limit or right side limit at every point'

HF: · 发表于 2006-10-30 19:38:54

I think it is true even for the weak condition that: 'at every point, either the left side limit or the right side limit exists'

constant · 发表于 2006-10-30 19:50:00

"result can be stronger"

yma16 · 发表于 2006-10-30 21:01:24

Please provide some details-eom

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