What is generally required to ensure that a strictly monotonically increasing function,is continuous and/or surjective?
F is an (injective) function F: $[0,1]\to [0,1]$ 2. F strictly monotonically increasing 3. F is a continuous' function on $[0,1]$ 4.. $F(1)=1$ and $F(0)=0$
To prove that F is sur-jective;and bijective function is derived on the basis of unique Global minima and global maxima, (where F(1)=1 and F(0)=0, are said maxima and minima, for example), set at the end points of both domain and co-domains via the extreme and intermediate value theorems?
Is this how sur-jectivity is often derived, that is via the extreme and intermediate value theorems using continuity and strict monotinicity?
Ie by the extreme value theorem, the function it will attain its maximum and minimum values in its range at those, and that by the Intermediate value theorem** the Function will attain all values in the co-domain between those two points.
Thus, if the maximum and minimum function values are equal in value to end points of the co-domain and domain respectly, and occur, only at the end points of the domain,(injective)? Is that correct
ie at F(1)=max, F(0)=min, then the function will attain every value between the f those, max and minimum points.
That is for every y in the co-domain $[0,1]$,such that yjust is [0,1] it will be sur-jective, but as it is in-jective (as F is strictly monotonic increasing) it will be bi-jective as well. Is that correct.?
ill a Midpoint Convex on f:[0,1] to [0,1] with F(0)=0 and F(1)=0; Strictly Increasing, non-negative, and Strictly Quasi convex and strictly quasi concave and concave, be continuous for is instance. It is bounded in some sense of the word? (that satifies the above constraints, but with continuity replaced by midpoint convexity, and F(0.5)=0.5