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example of differentiable function which is not continuously differentiable
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(Example)
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Let $f$ be defined in the following way:
Then if $x\neq 0$ , $f'(x)=2x\sin\left(\frac{1}{x}\right) - \cos\left(\frac{1}{x}\right)$ using the usual rules for calculating derivatives. If $x=0$ , we must compute the derivative by evaluating the limit$$ \lim_{\epsilon\to 0} \frac{f(\epsilon)-f(0)}{\epsilon}$$ which we can simplify to$$ \lim_{\epsilon\to 0} \, \epsilon\sin\left(\frac{1}{\epsilon}\right).$$ We know $\left|\sin(x)\right|\leq 1$ for every $x$ , so this limit is just $0$ . Combining this with our previous calculation, we see that
This is just a slightly modified version of the topologist's sine curve; in particular,$$ \lim_{x\to 0} f'(x)$$ diverges, so that $f'(x)$ is not continuous, even though it is defined for every real number. Put another way, $f$ is differentiable but not $C^1$ .
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"example of differentiable function which is not continuously differentiable" is owned by Koro. [ full author list (2) | owner history (1) ]
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| Keywords: |
topologist's sine curve |
This object's parent.
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Cross-references: differentiable, real number, even, continuous, diverges, topologist's sine curve, limit, derivatives
This is version 5 of example of differentiable function which is not continuously differentiable, born on 2004-02-19, modified 2004-08-02.
Object id is 5597, canonical name is ExampleOfDifferentiableFunctionWhoseDerivativeIsNotContinuous.
Accessed 6944 times total.
Classification:
| AMS MSC: | 57R35 (Manifolds and cell complexes :: Differential topology :: Differentiable mappings) | | | 26A24 (Real functions :: Functions of one variable :: Differentiation : general theory, generalized derivatives, mean-value theorems) |
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Pending Errata and Addenda
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