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When we were in primary school, teachers showed us graphs of ‘continuous’ functions and said something like
“Continuous functions are those you can draw without lifting your pen”
With this in my mind I remember thinking (something along the lines of)
“Oh, that must mean that if the function takes $2$ values $f(y)<f(z)$ then for every $c$ between $f(y), f(z)$ there must be some $x, (y<x<z)$ such that $f(x)=c$”
And that’s what I thought a continuous function was. But then the $epsilon$-$delta$ definition appeared, which put a more restrictive condition on what a continuous function was.
So, my question is, given the fact that Darboux functions “seem continuous” (in some subjective sense, I guess), why wasn’t this used as the definition of continuity? More generally, how did today’s (analytical) definition of continuity appeared?
You can’t actually draw many Darboux functions without lifting your pen from paper. The extreme example is the Conway base 13 function, which takes every value in every interval and hence is a Darboux function. You can’t however begin to even imagine what this function looks like, let alone draw it with a pen.
So you may want to require the function to be bounded. However this still fails, since you can get functions like,
$$ f(x) = begincases sin (1/x), & textif x neq 0 0 & textif x = 0 endcases $$
Which again, you can’t draw. So one idea may be to require functions $f:[a,b] rightarrow Bbb R$ to be of bounded variation, meaning that,
$$ supleftf(x_i)-f(x_i-1) < infty $$
Intuitively, this ‘measures’ the graph of the line $(x,f(x)) mid x in [a,b] $ and requires it to be finite. Then we intuitively should get a finite length curve which we can actually draw without lifting your pen.
It turns out however, that Darboux functions which have bounded variations are actually continuous. So in attempt to define continuity in a more intuitive way, we have found a more restrictive definition.
Even worse, this still isn’t enough. One can show that the function,
$$ f(x) = begincases x^3sin (1/x), & textif x neq 0 0 & textif x = 0 endcases $$
is continuous and has bounded variation, but you can’t really draw it. Note this function is also differentiable and has continuous first derivative. I’ll spare you the details, but using similar ideas we can also construct infinitely differentiable functions of bounded variation, which you can’t draw on paper.
From this, I think you can see why we don’t try to model continuity off the intuitive definition. Instead, we adopt the usual definition because it’s a much more useful and interesting class of functions to work with.
The Darboux definition does not correspond very well with our intuition about continuity. For example, the Conway function takes on every value in every interval, and is therefore Darboux. However it is not continuous, and I don’t think we want it to be continuous, because it certainly doesn’t agree with your teacher’s definition of drawing without lifting your pencil.
BINIBINING Pilipinas candidates with cancer patients at the Philippine Children’s Medical Center
This year’s batch of Binibining Pilipinas pageant hopefuls lived up to the catchphrase, “confidently beautiful with a heart,” made famous by 2015 Miss Universe Pia Alonzo Wurtzbach, when they spent time with cancer patients at the Philippine Children’s Medical Center in Quezon City last month.
All 40 aspirants took part in the “charity day” organized by pageant sponsor Dairy Queen, which saw them playing with the young patients, sharing stories with the parents, and distributing gifts and snacks to the underprivileged beneficiaries of the government-run medical facility.
Women from the medical field are among this year’s contestants.
Angela Lauren Fernando, a graduate of medicine from University of Santo Tomas, who is currently taking her internship at Makati Medical Center, said activities such as the one they participated in give the patients “happiness and hope.”
Fernando, a 24-year-old Kapampangan, also said patients’ parents also benefit from such visits. “They’re down, and this gives them the opportunity to see their children happy. When parents see them smiling, it also makes them happy.”
Fellow Kapampangan Roshiela Tobias, a 22-year-old operating room nurse, said “through our presence, we inspire them.”
Karen Ibasco, a 25-year-old medical physicist and teacher from Manila, said “activities like this brighten the patients’ day.”
“I joined Binibining Pilipinas to help patients like them. I see it as a platform to make people aware of sickness like this,” she said.
But the ladies said they see themselves benefiting more from the activity.
“It gives us the inspiration to be a model to other people. We also hear their story, fighting the struggle of a life-threatening disease we’re lucky not to have. So, I look back at my own struggles and tell myself, ‘I can do this. Because if they can handle their problems, so can I,’” said Fernando, who aspires to become a surgeon or pediatrician after her internship.
For Ibasco, visiting patients with cancer is a source of motivation to further pursue her passions.
“You get the opportunity to be grateful for what you have, to appreciate life,” said 2015 Miss Intercontinental first runner-up Christi Lynn McGarry, who also graced the affair. “It gives me hope because they are able to stay positive in the darkest situations.”
Holding charity activities, McGarry said, also uplifts the image of the pageant, by promoting the concerns of children and other sectors in need.
Get to know more about the 40 candidates in a special program, dubbed “The Road to the Crown” on ABS-CBN’s “Sunday’s Best” bloc on April 10 after “Gandang Gabi, Vice.”
The 2016 Binibining Pilipinas coronation night will be staged at Smart Araneta Coliseum on April 17, with a telecast on ABS-CBN’s “Sunday’s Best” bloc on the same evening.
Xian Lim and KC Concepcion will host the ceremony, together with 2014 Binibining Pilipinas first runner-up Laura Victoria Lehman as backstage correspondent.
A teacher in Newtown, Connecticut, where dozens of children were gunned down at the Sandy Hook Elementary School in 2012, was arrested on Wednesday morning for allegedly carrying a loaded gun to class.
The teacher was identified by police as Jason Adams. He is an 8th grade science teacher at Newtown Middle School, a school in the same district as Sandy Hook Elementary, according to local news outlet News Times.
Possession of a gun on school property is illegal and considered a class D felony under Connecticut state law. Read more…
More about Sandy Hook, Newtown, and World
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