Nowhere differentiable, continuous

[astronut24]

Hello
need help with this one.

f:[0,1] --> [0,1]

f( .x1 x2 x3 x4 x5 ...) = .x1 x3 x5 x7

( decimal expansion)

prove that f is nowhere diffrentiable but continuous.
i tried by just picking a point a in [0,1] and the basic definiton of differentiability about that point...doesnt seem to work! kindly help.

 

[jostpuur]

Let x be fixed. You have to show that the limit

$$\lim_{y\to x} \frac{f(y)-f(x)}{y-x}$$

does not exist. One way to show this is to choose two different sequences $$y_n\to x$$ for which the limit

$$\lim_{n\to\infty}\frac{f(y_n)-f(x)}{y_n-x}$$

is different. I think I succeeded in choosing two sequences so that one gave 0 as the limit, and other one 1 as the limit, but naturally I'm not throwing all the details here

 

[jostpuur]

Gladly I didn't show all the details, because my sequence that gave 1 as the limit, had mistakes in it.

EDIT: I think I got it dealed with. I'll return to the topic if the discussion goes to the details.