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Continuous Convergent Sequence Limit Real Number Differntiable Function Even Number Derivative Example Of A Connected Space Which Is Not Path Connected

1	Limit
1	Differntiable Function
1	Continuous
1	Real Number
1	Convergent Sequence
1	Even Number
2	Derivative
16	Example Of A Connected Space Which Is Not Path Connected

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Example Of Differentiable Function Whose Derivative Is Not Continuous

Let be defined in the following way:

$\displaystyle f(x)=\begin{cases} x^2\sin\left(\frac{1}{x}\right) & \text{if }x\neq 0\\ 0 & \text{if }x=0. \end{cases}$

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

, we must compute the derivative by evaluating the limit

$\displaystyle \lim_{\epsilon\to 0} \frac{f(\epsilon)-f(0)}{\epsilon}$

which we can simplify to

$\displaystyle \lim_{\epsilon\to 0} \, \epsilon\sin\left(\frac{1}{\epsilon}\right).$

We know $\left\vert\sin(x)\right\vert\leq 1$ for every

, so this limit is just 0. Combining this with our previous calculation, we see that

$\displaystyle f'(x)=\begin{cases} 2x\sin\left(\frac{1}{x}\right) - \cos\left(\frac{1}{x}\right) & \text{if }x\neq 0 \\ 0 & \text{if }x=0. \end{cases}$

This is just a slightly modified version of the topologist's sine curve; in particular,

$\displaystyle \lim_{x\to 0} f'(x)$

diverges, so that

is not continuous, even though it is defined for every real number. Put another way,

is differentiable but not