The Topologist’s Sine Curve.

The topologist’s sine curve.

Limts: f(x)=sin(1/x) is a rare example of a function with a non-existent one-sided limit. More technically, f(x)=sin(1/x) is defined for all numbers greater than zero, yet the limit as x approaches zero from the right of f(x)=sin(1/x) does not exist. This can be reasoned by considering the value of f at x-values near zero. Informally, f(near zero) could be 1 f(just a bit closer to zero) could be -1 so f(numbers near zero) does not seem to settle on a single y-value.

Continuity: Note that f is continuous for all numbers greater than zero but not continuous at x=0 since f is undefined there. Even if we were to “fill in the bad point” and let f(0)=0, the function would still not be continuous at zero! (note this is the natural choice as sin(0)=0). We can see that the adjusted f is still not continuous at zero since the sequence x_n=1/(pi/2+npi) converges but f(1/x_n) is the sequence (-1)^n which does not converge. This is similar to the argument above. In other words, closing in on x=0, we can keep finding x values such that f(x)=-1 and f(x)=1.

Topology: In topology, the topologist’s sine curve is a classic example of a space that is connected but not path connected. This space is formed in R^2 by taking the graph of f(x)=sin(1/x) together with its limit points (the line segment on the y-axis [-1,1], the red line on the second image). The graph of f is connected to this line segment as f and the segment cannot be sepearted by an open disc (no matter how small). This can be informally reasoned by the zooming illustration in the second image. But the space is not path connected by the sequence argument above (there is no path to the point (0,0)).

Image credits: http://mathworld.wolfram.com/TopologistsSineCurve.html and https://simomaths.wordpress.com/2013/03/10/topology-locally-connected-and-locally-path-connected-spaces/

for all of you guys getting anxious about the environment and pollution

y’all deserve to hear the good news too, and yes i fact checked these.

7 miles of habitat for bees will be planted in london

the biggest coal plant in north america has been converted to solar panels

roads in edinburgh will close once a month to help pollution

maine has banned styrofoam

new york city and los angeles have both made “green new deals”

The london marathon replaced 200,000 single use water bottles with seaweed water pods

the bees in notre dame survived the fire, and the roof could possibly built in a more eco-friendly way.

the population of flightless kakapos (a cool bird) is rising

the carbon emissions in the uk are the lowest they’ve been since 1998

Another large bee habitat (.5 mil acres) has been created

there is a robot that delivers young coral to help repopulate the great barrier reef

portugal plans to stop using plastic on fruit, vegetables, and bread by 2020

106 new species of bees have been discovered in australia since 2010

a group of Sikhs plan to plant 1,000,000 trees as a gift to the earth

disney has made a mickey-shaped solar farm

Morgan freeman turned a 124 acre ranch in mississippi to a bee habitat

China plans on making a “forest city” to help clean up their air

An increasing amount of countries and states are beginning to ban single use plastics.

A couple replanted a whole forest in brazil (2.7 mil trees) in 20 years, and the animals have come back to live there

The hole in the ozone layer is repairing itself more each year

China plans on spending 360 billion dollars to improve renewable energy and has scrapped plans for coal powered plants that were going to be built.

A national park has been built in the amazon (3.3 mil acres in peru) to preserve the rainforest

Ireland and the uk have declared a climate emergency

The guy who played aquaman (jason momoa) has spent 31,000 dollars to help clean up plastic

South korea is now recycling 95% of food waste

There is a cleanup campaign being planned for mt. everest

Puerto rico wants to use all renewable energy by 2050 and is setting official goals for that

Some schools have special water bottle fill-up stations that encourage you to use refill and reuse plastic bottles rather than throwing them out after one use, these are becoming more common.

9 endangered species are thought to make a comeback this year

20 countries in africa are planning to make a “green wall” of trees and plants that will span the width of africa to stop desertification

recently, a lot of volunteers and organizations are planting a ton of trees.

Awareness about the environment and climate change is growing super fast right now among people and countries, which will only help us

Most pollution is caused by like 100 companies, but a few of those (like pepsi) are trying to cut down on that.

if this stuff keeps happening things will get even better, and the only thing stopping us really are those big companies who don’t want to pay the money to switch to more eco-friendly energy sources. i know the media tends to cause fear and stress about this stuff for a lot of people, but there really is hope.

easy things we can do:

Recycle

Plant gardens in your yard for bees if you can

Participate in community volunteer things that plant trees and gardens

Theres this search engine called ecosia that plants a tree for every 45 searches you make, it has almost 2 mil users.

Pick up trash if you see it when you’re at the beach or in nature

If you have a fair amount of money, consider donating some to trustworthy environmental organizations

Start using a reusable water bottle (like those ones at target) rather than relying on single use plastic ones.

Cut the plastic rings on plastic milk bottles. You know, the little spiky plastic ring near the cap. Birds get those things stuck around their neck and die, so cut them so that they can’t get stuck on a neck.

If you can afford it, get some of those reusable grocery bags and sue those. most grocery stores have them, and it saves a lot of waste.

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chaos3612

6 years ago