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

spring-of-mathematics:

isomorphismes:

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. Letlim 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.

spring-of-mathematics:

isomorphismes:

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]

image

Also, If (m,n) ⊂ (a,b), the function f:(a,b)→ℝ, that is also true.

(Source: talizmatik)

thelovelyseas:

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

thebrainscoop:

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!" 

thelovelyseas:

Manatee (x)

thelovelyseas:

Manatee (x)

lunarspace:

Ooft

lunarspace:

Ooft

(Source: lifeless-queen)

bile4:

Go off

(Source: archiemcphee)

future-robin:

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)

youre-my-snowflake:

pnbawesome:

Wilis in Giselle.


Holy shit this makes me want to cry

youre-my-snowflake:

pnbawesome:

Wilis in Giselle.

Holy shit this makes me want to cry

collegehumor:

This HR dept doesn’t negotiate with Terrorists.

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