Let [itex]f: X \rightarrow \mathbb{R}[/itex]. The definition given for upper semicontinuity at a point [itex]a[/itex] is for any positive [itex]\epsilon[/itex] there is a positive [itex]\delta[/itex] such that if [itex]|x - a| < \delta[/itex] then [itex]f(x) < f(a) + \epsilon[/itex].(adsbygoogle = window.adsbygoogle || []).push({});

My understanding is, for ordinary continuity at the point [itex]a[/itex] then the inequality [itex] |f(x) - f(a)| < \epsilon [/itex] or [itex] f(a) - \epsilon < f(x) < f(a) + \epsilon [/itex] holds. Therefore for upper semicontinuity we are interested in the right part of the inequality (for lower semicontinuity, we are interested in the left part).

For example the graph

[tex]f(x) = \left \{ \begin{array}{cc} x^2, & x \neq 1, \\ 2, & x = 1 \end{array} \right .[/tex]

is upper semicontinuous at [itex] x = 1 [/itex]. Does that mean that upper (lower) semicontinuity relaxes the requirement that a jump may occur up (down) at that point?

1. What is the story behind it?

2. Is there any real life example or analogy that we can relate to of this concept? (How to explain it to a precocious eight year old child?)

3. Why do we need the notion of upper (lower) semicontinuity?

4. What differentiate the notion of upper (lower) semicontinuity from that of left/right limit?

4. In what context is the notion of upper (lower) semicontinuity used widely?

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# Semicontinuity for single variable function. What is it all about?

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