Zero derivative implies locally constant: Difference between revisions
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Suppose <math>f</math> is a [[function]] and <math>I</math> is an [[interval]] (possibly open, closed, or half-open and half-closed) contained inside the domain of <math>f</math>such that <math>f'(x) = 0</math> for all <math>x</math> in the interior of <math>I</math> and <math>f</math> is ''continuous'' on all of <math>I</math> (''note that we do not require differentiability, or even one-sided differentiability, at the endpoints of <math>I</math>, if any''). Then, <math>f</math> is a constant function on all of <math>I</math>, i.e. there is a real number <math>C</math> such that <math>f(x) = C</math> for all <math>x \in I</math>. | Suppose <math>f</math> is a [[function]] and <math>I</math> is an [[interval]] (possibly open, closed, or half-open and half-closed) contained inside the domain of <math>f</math>such that <math>f'(x) = 0</math> for all <math>x</math> in the interior of <math>I</math> and <math>f</math> is ''continuous'' on all of <math>I</math> (''note that we do not require differentiability, or even one-sided differentiability, at the endpoints of <math>I</math>, if any''). Then, <math>f</math> is a constant function on all of <math>I</math>, i.e. there is a real number <math>C</math> such that <math>f(x) = C</math> for all <math>x \in I</math>. | ||
==Facts used== | |||
# [[uses::Lagrange mean value theorem]] | |||
Latest revision as of 20:11, 7 September 2011
Statement
For an interval in the domain
Suppose is a function and is an interval (possibly open, closed, or half-open and half-closed) contained inside the domain of such that for all in the interior of and is continuous on all of (note that we do not require differentiability, or even one-sided differentiability, at the endpoints of , if any). Then, is a constant function on all of , i.e. there is a real number such that for all .
