Theorem: A set in R is compact if and only if it is closed and bounded. wwwwwwwwwww A) Determine and justify whether the set [-1, 1] is compact or not. B) Justify because the function f(x)=x^2 is continuous, also find the image of [-1, 1] under f, wwwwwwwwwwwwwwwwmmmmm wwwwwwww that is, f([-1, 1]), justify because f([-1, 1]) is compact.

What is Compact Set:

Compactness is a quality used in mathematics, specifically general topology, to more precisely define the idea of a space having no gaps or missing endpoints, i.e., that the space does not exclude any limiting values of points. A topological space is said to be sequentially compact if every infinite sequence of points sampled from the space has an endless sub-sequence that converges to some point in the space, according to one such generalisation. A subset of Euclidean space is said to be compact in this sequential meaning by the Bolzano-Weierstrass theorem if and only if it is closed and bounded.

Given:

Given set is $\left[-1,1\right]$. Also, a function $f\left(x\right)=x^2$ is given. 

To Determine:

We verify whether the given set is compact or not. We also verify whether the $f$ image of $\left[-1,1\right]$ is compact or not.