1617-257/TUT-R-5: Difference between revisions
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On 10/13/16, we proved that if <math>U</math> is an open and convex subset of <math>\mathbb R^n</math> and |
On 10/13/16, we proved that if <math>U</math> is an open and convex subset of <math>\mathbb R^n</math> and |
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if <math>f : U \to \mathbb R</math> is differentiable with <math>\|D f (x)\| \leq M</math> for all < |
if <math>f : U \to \mathbb R</math> is differentiable with <math>\|D f (x)\| \leq M</math> for all <math>x \in U</math> |
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then we have that <math>|f(x) - f(y)| \leq M \|x - y\|</math> for all <math>x, y \in U</math>. |
then we have that <math>|f(x) - f(y)| \leq M \|x - y\|</math> for all <math>x, y \in U</math>. |
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Latest revision as of 10:34, 14 October 2016
On 10/13/16, we proved that if is an open and convex subset of and if is differentiable with for all then we have that for all .
We also proved the analogous statement if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} is Lipschitz continuous instead of having uniformly bounded derivative.
Lastly, we created a formulation for the problem if is star-shaped rather than convex.
