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?
If you want to take a function that is continuous over some interval
Suppose that a function is continuous over the closed interval
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