literally i was crying a few minutes ago because i couldn't do my analysis exercise sheet. You are the best teacher on youtube thank you so much, may god bless you have a beautiful day
I love that this video gives an intuitive explanation of the difference. Of course having the difference explained using the definition is important since it’s important to know that in uniform continuity, delta does not depend on the point and only depends on epsilon. However, this difference on the surface may seem unintuitive and unimportant. However, the rectangle demonstration perfectly encapsulates the intuitive idea behind why we care about uniformly continuous functions. For general continuous functions, there exists a rectangle that encloses the value of the function over some delta interval of x values, but the height of that box (and thus the area of the box) may depend on the specific point of the function that we are looking at. A given delta interval of x values does not always correspond to the same epsilon interval of function values due to the influence of the point itself. Uniformly continuous functions for a given delta interval for x have the same rectangle at every point of the function. The height of the rectangle, the epsilon interval for the image of the function, only depends on the given delta value. Hence the name uniform continuity.
I just finished my (intro to) Real Analysis course for my Applied Math major, and uniform continuity was the last topic in the class. Thanks for helping me get an A!
Thank you brother. I just realised i passed through Real analysis without intuitively understanding this. Sometimes we can just continually talk about the definition without intuitively understanding it. This intuition is the best. Continue the good work.
Thank you so much!! I'm studing real analysis this semester and it all seems so theoritical, it's hard to draw a picture like that. Your videos really help with understanding the material, thanks again.
I think a better way to understand Uniform Continuity (not sure if it's accurate) is that if a function is uniformly continuous on the real set of numbers, then the function is continuous and the slope of the tangent line of every point of the function is within a finite range.
Pretty good video, but it seems like you call f(x) values “y values” which can be confusing given the definition. The definition states that both x and y lie in the domain of f. What that means for the definition is that you can think of x and y as 2 different “x values” (by x values I mean domain values, as in they are defined by their position on the x axis). For uniform continuity, we see that the difference between 2 points is arbitrarily small (less than delta) and thus the difference between f(x) and f(y) is also small (less than epsilon). By small, I mean small enough to satisfy the definition of uniform continuity. The rectangle visualization is still correct though. When an f value (escapes) the rectangle, that would mean that abs(f(x)-f(y)) is no longer less than epsilon and is therefore not uniformly continuous at the point (y,f(y)). I hope what I said can help anyone who might’ve been confused by that. If it was just me who was confused than oh well.
The Math Sorcerer yeah man I was mostly just putting my thoughts here so I could better understand what’s going on. Appreciate the instant feedback to my comment as well that’s nice. Keep up the good work!
Thanks for this video. I found it really helpful. I completed my masters this year. But I still get confused between uniform continuity and continuity😭😭. Finally the confusion is gone. Thanks a lot.
Hello Sir...If a function is uniformly continuous on a closed interval, could we refine the definition of uniform continuity by replacing the condition |x-y| < δ and |f(x) - f(y)| < ε with |x-y| ≤ δ implying |f(x) - f(y)| ≤ ε ? Please sir, clarify it.
Sir, If a function is uniformly continuous on a closed interval, could we refine the definition of uniform continuity by replacing the condition |x-y| < δ and |f(x) - f(y)| < ε with |x-y| ≤ δ implying |f(x) - f(y)| ≤ ε ?
Very understandable 😍 But one point is making me confuse In the very last when u talk about rectangles ; in uniform continuity the size of rectangle should same at one place of graph when compared with continuity or at any point of graph size of rectangle should be same.?
hi I have some comments: firstly, thanks for the video and explanation ;) second, you should work the voice for next videos some parts it's very low then goes up suddenly. the most important comment is about uniform continuity I didn't get the drawing of a rectangle what if it goes like a wave but not down. when can we decide whether it goes away from the rectangle or not? I like that you have explained analytically and geometrically !! thanks again
In your example with the red function near the end of uniform continuity, why could we not change the rectange to be taller so that we capture all of f(x) and f(y)? If it the rectangle is tall enough for the "worst case", shouldn't it be tall enough for the whole function and then be uniform continuous?
I don't see a difference in the definitions except for the order they're written. It says for continuous, it's dependent on c but in uniformly continuous it's dependent on y, but y and c are both just real numbers, you just changed the name of the variables, they're still just in R doesn't matter what you call them, so no difference there.
I was confused with this as well... but then I noticed what he said: that Delta in the Uniform Continuity can only depend on Epsilon, not on x. The language used to define this is specific, as follows: Whereas the first definition mentions the existence of "c" before it mentions that Delta exists, the second definition does not mention X or Y before it mentions that Delta exists. It's a subtle difference in linguistics, but is intentional. It's the mathematicians way of saying that Delta can depend on "C" in the first definition but cannot depend on X or Y in the second.
In case of continuity delta depends on c so - that value is constant. While in case of uniform con.. delta depends only on epsilon so no matter what c is that's why that is choosed like a variable y, we can take any no. Of our choices In place of y.
It's very different, in Continuity definition, the "c" is constant, that means that just "x" changes in |x - c| < d, in another terminology, it's a neighborhood with radius 'd' and center 'c' B_d(c), and if 'x' belongs to that neighborhood, then ocurrs that |f(x) - f(c)| < e. or in other terminology, f(x) belongs to B_e(f(c)). In Uniformly contininuity, we have that |x - y| < d, both changes, it can be any number whose distance is less than 'd'. In ths case, there are not center, not neighborhood, just distane between two numbers
Still don't understand the uniform continuity part :/ Guess I've got a low IQ. But I'll think about this visual intuition for some time and hopefully I get it
Random Dude Natural language helps me: A function is continuous means... Choose any input value for the function. Whatever you like. And choose any distance epsilon. Then I can find you a distance delta such that just so long as I keep my input values within delta *of the input you chose* , I’ll keep the correponding outputs within epsilon. A function is uniformly continuous means... Choose any distance epsilon. Then I can find you a distance delta such that just so long as I keep *any pair* of input values within delta, I’ll keep the outputs within epsilon. Where does the difference emerge? In the case of continuity, depending on which input you choose, I might need to keep my other input values within *different* distances (some may need to be really small, I might be safe keeping some fairly large) to keep their outputs within epsilon. Whereas in the case of uniform continuity, no matter what input you choose, I can always find the *same* distance delta between other input values which keeps their outputs within epsilon.
Go through the video ( URL ) given below for best explanation for the difference between continous function and uniform continous function: ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-hXkQqCBLRp8.html
