Give examples without justification An open cover of 0infini

Give examples without justification. An open cover of (0,infinite) that has no finite subcover. A function g:(0,l) (0,1) that is continuous at every point except x = 1/2

Solution

Consider the set (-1,0]U(-2,0]U(-3,0] upto infinity

Clearly this is not a finite set as n becomes unduly large.

It is also open

And this open cover does not have a finite subcover.

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f:g(0,1) -->(0,1)

Consider this function f(x) = 0, 0<x<1/2

= 1/2, 1/2 <x<1

This function being constant is continuous in (0,1/2) U(1/2,1)

But not continuous at x =1/2 as limits are different.