How do I prove/disprove " if a function is differentiable, then it is Lipschitz" ?
2007-01-02 12:13:59 · 11 answers · asked by jackie 1 in Science & Mathematics ➔ Mathematics
Differentiation expresses the rate at which a quantity, y, changes with respect to the change in another quantity, x, on which it has a functional relationship. Using the symbol Δ (Delta) to refer to change in a quantity, this rate is defined as a limit of difference quotients
which means the limit as Δx approaches 0. In Leibniz's notation for derivatives, the derivative of y with respect to x is written
suggesting the ratio of two infinitesimal quantities. The above expression is pronounced in various ways such as "d y by d x" or "d y over d x". The form "d y d x" is also used conversationally, although it may be confused with the notation for element of area.
Modern mathematicians do not bother with "dependent quantities", but simply state that differentiation is a mathematical operation on functions. One precise way to define the derivative is as a limit [2]:
A function is differentiable at a point x if the above limit exists (as a finite real number) at that point. A function is differentiable on an interval if it is differentiable at every point within the interval.
2007-01-02 12:35:05 · answer #1 · answered by kellenraid 6 · 0⤊ 0⤋
Probably by finding a function that is differentiable but not Lipschitz. For example, the function f(x)=x^2 on the real line. However, if the derivative is bounded the function is Lipschitz by the Mean value Theorem.
There are functions with unbounded derivative that are still Lipschitz (like f(x)=sqrt(x)).
2007-01-02 20:29:16 · answer #2 · answered by mathematician 7 · 1⤊ 1⤋
It's false. Here's a counterexample from Wikipedia:
The function f(x) = x² with domain all real numbers is not Lipschitz continuous. This function becomes arbitrarily steep as x ââ.
To have Lipschitz continuity you need to have a bounded
first derivative.
2007-01-02 20:27:36 · answer #3 · answered by steiner1745 7 · 0⤊ 1⤋
Huh?!
You need to read the text book about this. It is a very well documented feature. Look it up on google, but essentially it's, in part, the condition where the function is continuous over the interval being differentiated.
2007-01-02 20:18:39 · answer #4 · answered by Anonymous · 0⤊ 0⤋
Can you Ping in a Segmented Network that is Independent?
Thanks, RR
2007-01-02 20:17:40 · answer #5 · answered by Anonymous · 0⤊ 2⤋
I prefer abstract art; you may too. GL on the math thing.
2007-01-02 20:16:32 · answer #6 · answered by Hushyanoize 5 · 0⤊ 1⤋
Is lipchitz not a group of rabbis in poland
?
2007-01-02 20:16:59 · answer #7 · answered by greek302 2 · 0⤊ 2⤋
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2007-01-02 20:18:59 · answer #8 · answered by christopher_az 2 · 0⤊ 0⤋
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2007-01-02 20:16:50 · answer #9 · answered by Anonymous · 1⤊ 2⤋
add them together
2007-01-02 20:15:46 · answer #10 · answered by yo 1 · 1⤊ 3⤋