Please Subscribe here, thank you!!! goo.gl/JQ8Nys How to Prove a Function is Uniformly Continuous. This is a proof that f(x) = 1/(1 + x^2) is uniformly continuous on R. I hope this video helps.
Since 1/|1+x^2| is bounded above by 1 and |x|/|1+x^2| is bounded above by 1/2 (and similarly for y), will the proof still hold if we had chosen delta = epsilon?
mr math sorcerer, at 7:30 can we prove this segment by proving the bottom part of the fraction is bigger or equal to the top part? what i mean by this is by plugging in any number for 'a' we get a result smaller or equal to one
Why is the last inequality "less than or equal to 1" instead of strictly less than 1? What I mean is, does there exists a number a such that (|a| / |1+a^2| )= 1?
In the proof at the end, shouldn't you be considering the cases where |a| 1? The way you did it means that if a < -1, then |a| > 1 and you can't use the first inequality in case 1.
Matt Hutchings oh yeah! I get your point. Those cases work only for non-negative numbers. For the whole real line we have to go with cases where a is between -1 & 1 and otherwise. Great observation.
Well, since it is absolute value it must be positive. And of if it is positive it can only makes that whole fraction smaller number. So if you get rid of it it doesnt really matter.
Ivan Ivan It indeed becomes a smaller number, but we claim that is is greater of equal, right? So how can two numbers suddenly be greater when they become smaller?
could just let delta equal n* epsilon at first then when he got to the end of proof just change n to 1/2 as that clearly works nicely with the mod(f(x)-f(y)) being less than epsilon
immediately frustrated because you figured out delta beforehand and was just like ok there you go. now let's do more redundant stuff to see if it's actually what I put down though I already told you it is.
Exactly. Finding the delta is the actual problem. Not showing that the chosen delta works. I fuckin despise these delta epsilon proofs, they make me go insane.
I'm confused. Why is it redundant...? You won't be given delta in an actual problem.. He just said, "this is what I came up with," and showed us how he came up with it.
I am sooo late I have a doubt...i hope you reply.....i understand the proofs...but is there any tips on determining the delta......i understand it depends on the question but some tips on that
Great video but please explain how is this function uniformly continuous if when I graph it it does not have a constant slope (isn't that required to be uniformly continuous)
+James Digno a^2>=0 for all a in R, so (1+a^2)>=(1) => mod(1+a^2)>=mod(1) thus dividing by the smaller expression will give a greater expression as it is a reciprocal (laws of inequalities). A similiar argument will convince you that dropping the 1 will also give the same result.
The absolute value of the difference of real numbers is a metric on the real numbers, so you may change the order of the terms and obtain an equivalent expression. More simply, notice that the absolute value of a difference gives the distance on a number line between the two values. This distance shouldn't change just because we change the order of the values. Ex. 4 - 9 = -5, 9 - 4 = 5, but |-5|=|5| = 5
because it's absolute value we know the things he is "dropping" are positive and we also know that the bigger the denominator of a number the smaller the number. So if we're working with inequalities it follows that |x|/ |1+x^2| |1+y^2| < |x|/ |1+x^2|
+The Math Sorcerer um if memory serves the rule I worked out was to check what is the largest number the 1/(1+x^2)+1/(1+y^2) could be which is 2 in this case the dive epsilon by that number and call it delta.
I still can't see the difference between continuity and uniform continuity. I think I don't understand them correctly. Please please please please please make video on them and show the differences
