Herr Lipschitz: Continuously Yours!

Basic calculus: Continuity: Let A ⊂ R and f : A → R be continuous. Then for each x0 ∈ A and for given ε > 0, there exists a δ(ε, x0) > 0 such that x ∈ A and | x − x0 |< δ imply | f(x) − f(x0) |< ε. (Observe that δ depends, in general, on ε as well as the point x0).

A little more advanced: Uniform Continuity (Definition): A function f : A → R, where A ⊂ R is said to be uniformly continuous on A if given ε > 0, there exists δ > 0 such that whenever x, y ∈ A and | x − y |< δ, we have | f(x) − f(y) |< ε.

Question 1: What is the difference between continuity and uniform continuity? Which is the stronger condition? Why?

Once you've clearly resolved the above and looked at a few examples to see the difference clearly (the internet is a great resource if you know where to look!):

Question 2 (Self Study): What is Lipschitz Continuity? Is it a stronger or weaker condition that continuity and uniform continuity?

After you've resolved the above at the level of definition and theory, make sure you look at few examples and solve a few problems so that the concepts settle in your mind clearly.

PS: Even if you're not a mathematician, if you're using functions, differential equations, etc., you ought to be curious about what you're using and why it works!

Comments

Post a Comment

Popular posts from this blog

Irrationally yours, e!

Alright, here's some fun (and charming!) mathematics for a peaceful Sunday afternoon: Using the infinite series representation of e^x = 1 + x + x^2/2! + x^3/3! + ... + x^n/n! + ..., prove that e must be irrational. (Hint: Go after 1/e. Trap this minion and he shall lead you to the Don!) Note: No, e is not irrational by definition! (Yes, of course, if you are stuck after trying for a while, by all means consult references (or reliable sites on the internet). I'm simply putting up math problems that I find charming 😊. These problems have made me appreciate how beautiful mathematics can be if you are a bit patient.)
Read more

Unique Factorization Theorem

Alright, let's start having some maths fun with this blog 😉! This first post ought to be bread and butter for math and computer science majors, but I think it's something pretty cool for the intellectually curious anyway! Consider this : Every integer n > 1 can be represented as a product of prime factors in only one way, apart from the order of the factors. That's crisp! Isn't it? The above is known as the Unique Factorization Theorem. (It's also called the Fundamental Theorem of Arithmetic - that's how important this result is considered.) Let me know using an appropriate reaction emoticon in the comments fields below if you can prove this theorem. If not right away, then take your time - no hurries 😉!
Read more

The Neighborhood of Infinity

Image
Did you know that: 1) Although the more commonly used geometric representation of complex numbers is the complex plane (developed by Gauss (1799) and Argand (1806)), one can also use the sphere (Riemann (19th century)) to obtain what is known as a stereographic projection (see figures below). Here points, P, of the sphere are projected from the North Pole, N, onto the tangent plane at the South Pole as follows: The line passing through N and P intersects the tangent plane at the point P'. This P' represents the sphere point P in the plane. Observe that with the exception of the North Pole itself, each point of the sphere corresponds to exactly one point in the plane. (Wait...there's coolness coming...keep reading...) 2. While we need two symbols '+infinity' and '-infinity' when we extend the real line R to infinity (the extended real line is called R*), we need only one symbol 'infinity' when extending the complex plane C to C*. This results form the...
Read more