Calculus Interlude - The Weierstrass or Extreme Value Theorem

This theorem states that a function that is continuous over a closed interval admits extrema and takes on all values inside that interval.

Remember how we said that continuity is essentially the ability for a function to have a "non-interrupted" or "consistent" graph, in very simple words?

It is exactly because of that.

If you want to take a function that is continuous over some interval , because of this continuity it absolutely has to have every value between a and b as an input.

"Okay, but what do we do with this information?"

Suppose that a function is continuous over the closed interval , and we know that:

Because of continuity, this function must take on all values of x in the interval .
However, the image of this function, so, the output, must be between these values of the output interval .

This can be a very useful tool in function analysis.
The Mean Value (Lagrange) Theorem