Cownose ray, Blue spotted ribbontail ray, Eagle ray, Bat ray, Southern Stingray...

Monotone and antitone functions

(not over ℝ just the domain you see =

`0<x<1⊂ℝ`

)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.,x→a+ x→b−lim f(x) = β≤+∞ and limf(x ) = α≥-∞, if f is strictly decreasing,x→a+x→b−

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.,y→α+ y→β−lim(f^{-1})(y)=b and lim(f^{-1})(y)=a, if f is strictly decreasing.y→α+ y→β−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.

(Source: talizmatik)

Hannah Fraser swims dressed as a mermaid with humpback whales off Vava’u Island, Tonga, to raise awareness of marine life and oppose whale hunting. Photos by Ted Grambeau

Only good things can come of this.

"Would you care for a date?"

"You mean…. another data point?"

"No, I… was thinking about dinner."

—-

"Joan, will you….

"Yes?"

"…be my co-author?"

"Oh, Meredith! Et al!"

This is honestly one of my favorite Spongebob moments. Can we take a minute to realize how clever the writing for this show used to be?

(Source: iraffiruse)

This HR dept doesn’t negotiate with Terrorists.

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This Is The Most Passive-Agressive Office Note Battle We’ve Ever Seen