Example of function with given condition

Question

Let $f$ is a differentiable function at a and {$x_n$} and {$y_n$} converging to a and $x_n \neq a$ ,$y_n \neq a$ and $x_n \neq y_n$. and I wanted to find example such that $$\lim_{n \to \infty}\frac{f(x_n)-f(y_n)}{x_n-y_n} $$such that
a) limit exist but not equal to $f'(a)$
b) limit does not exist.
I know that above limit equal to derivative if $x_n < a < y_n $ or vicersa .But I am not able to find example .Any Help will be appreciated .

Answer 1

Take a function whose differential is not continuous and cook an example.

Discontinuous derivative.

For example the Mark Mclure example here.

$f(x)=x^2sin({1\over x})$, $f'(x)=0$ if $x=0$ otherwise $f'(x)=2xsin({1\over x})-cos({1\over x})$.

Let $x_n,y_n\in [{1\over{2n\pi+\pi/4}},{1\over{2n\pi}}]$ distinct,

${{f(x_n)-f(y_n)}\over{x_n-y_n}}=f'(c_n)(x_n-y_n)=(2c_nsin({1\over c_n})-cos({1\over c_n}))(x_n-y_n)$, but since $c_n\in [{1\over{2n\pi+\pi/4}},{1\over{2n\pi}}]$, $cos({1\over c_n})\in [{\sqrt2\over 2},1]$. You deduce that $lim_nf`(c_n)$ is not zero.