Is there any continuous function having local minima at c but is neither decreasing in (c-h,c) nor increasing in (c,c+h)?

Question

I can find a discontinuous function having the above conditions..... f(x) =x for x lying betwen (0,1) and (1,2) and -10 at x=1 But can anyone help me in finding a continuous function satisfying the conditions....

What about $y=x^2\sin^2\frac{1}{x}$ for $x\ne 0$ and $y=0$ for $x=0$? — velut luna, Aug 29 '16 at 13:57

velut luna · Accepted Answer · 2016-08-29T14:36:27.167

3

I think you can consider the function $$y=\bigg\{ \begin{array}{ccc} x^2 \sin^2\frac{1}{x}& &x\ne0 \\0 && x=0\end{array}$$ It is continuous and $(0,0)$ is a local minimum, but does not satisfy your conditions.

edited Aug 29 '16 at 14:36

answered Aug 29 '16 at 14:02

velut luna

9,961

I still have a doubt...f(x) at 0.1 and - 0.1 are equal......i.e. it is decreasing in left interval and increasing in right..... – user364168 Aug 29 '16 at 14:40
@user364168 the values at some arbitrarily chosen points $\pm 0,1$ are of no relevance in case of a highly oscillating map like the one proposed here. – Thomas Aug 29 '16 at 15:07
Okay.....i understood....thank u all...... – user364168 Aug 29 '16 at 15:26

Is there any continuous function having local minima at c but is neither decreasing in (c-h,c) nor increasing in (c,c+h)?

1 Answers1