Real Analysis Live - Problem Solving - Continuous Functions

Together with @fabiangabel

00:00 Intro

[TheMrAineas]

Weierstrass min max theorem says 0<=m<=f(x)<=M<=1. g=f-x is continuous and g(m)g(M)<=0. Therefore there exists a point x0 in [m, M] for which g(x0)=0

[ThemJazzyBeats]

For problem 3, don't you need f to also be surjective ?
Or was it implicit that the interval [0, 1] is not only the codomain but the image (or range) of f ?

For instance, if f is not surjective, then wouldn't it just be able to "stop" at where the fixed point would occur, and since its not surjective, it would not need to "start again", which would make it not continuous ?

[FraserIland]

17:18
ERROR : the 4th degree parabola does non have any minimum in x = -2 but it is not important for the exercise. No impact on the solution