Paul Erdős, the most prolific mathematician in history, always made jokes about his age. He said for instance that he is 2.5 billion years old, because in his youth the age of the Earth was said to be 2 billion years old and around 1970, it was known to be 4.5 billion years.
This is such a teacher joke
- Male Teacher: Man, I wish there were just a guy who was just like, the World Master of Geometry - just call him up and he could provide an answer for any geometry question!
- Female Student: Who says they would have to be a guy?
- Me: YAAAAAAAAAAS
Monotone and antitone functions
(not over ℝ just the domain you see =
These are examples of invertible functions.
Theorem on the inverse function of continuous strictly monotonic functions:
Suppose the function f:(a,b)→ℝ with -∞≤a<b≤+∞ is strictly increasing (resp., decreasing) and continuous. Let
lim f(x)=α≥-∞ and lim f(x)=β≤+∞ , if f is strictly increasing, resp.,
lim f(x) = β≤+∞ and limf(x ) = α≥-∞, if f is strictly decreasing,
Then f maps the interval (a,b) invertibly onto the interval (α,β). The inverse function f -1:(α,β)→(a,b) is also strictly increasing (resp., decreasing) and continuous, and we have:
lim(f -1)(y)=a and lim(f -1)(y)=b, if f is strictly increasing, resp.,
lim(f -1)(y)=b and lim(f -1 )(y)=a, if f is strictly decreasing.
Analogous statements hold for semi-open or closed intervals [a,b].[Source]
Also, If (m,n) ⊂ (a,b), the function f:(a,b)→ℝ, that is also true.