Meital Posted September 28, 2005 Share Posted September 28, 2005 I am reading ODE ( Ordinary DE) notes, and there is a statement that says " A function that is not uniform, even a continuous function that is not uniform, cannot have a lipschitz constant. As an example is the function 1/x on the open interval (0,1). I want to see if I understood this correctly. I will assume that we can find a number k >= 0 such that | f(x) - f(y) | =< K*|x - y| we have x and y in (0,1) so 0 < x < 1 0 < y < 1 .....(1) Now, | 1/x - 1/y | =< k*|x - y| Find common denominator | y - x|/|xy| =< k*|x-y| | y - x | = |x - y| then we cancel the term from both sides of inequality, so we get 1/|xy| =< k ...($) k >= 0, and from (1) 1/xy > 1 so is 1/|xy| > 1 but then the equation ( $ ) becomes 1 < 1/|xy| =< k , but k >= 0 so this is contradiction? I am not sure, can someone tell me why 1/x doesn't have a lipschitz constant? I mean since we are in (0,1) then there is no upper bound for 1/|xy| so we can't have an upper bound for it (lipschitz constant) Link to comment Share on other sites More sharing options...
Dave Posted September 30, 2005 Share Posted September 30, 2005 That's the idea. I think the argument I saw a long time ago said that basically no matter which k we pick, we can always find x or y big enough so that 1/|xy| > k, so there could never exist a k big enough. Link to comment Share on other sites More sharing options...
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