Math Is Fun Forum

Fistfiz

Member

Registered: 2012-07-20

Posts: 33

Little derivative problem (prove or give a counterexample)

Hi guys, i'm trying to figure out if this is true or not, can you help me?

Conjecture: Let f:[m,+∞)->R be a continuous and monotonous function with a horizontal asymptote y0 (as x->+∞). Then:
1) f is derivable.
2) f'->0 as x->∞.

I ask for f being monotonous because the only counterexamples, to the non-improved conjecture, that came to my mind are things like f(x)=sin(x^2)/x.

Thanks in advance.

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30+2=28 (Mom's identity)

scientia

Member

Registered: 2009-11-13

Posts: 224

Re: Little derivative problem (prove or give a counterexample)

Not true. ƒ may only be piecewise differentiable, not wholly differentiable.

Let us define a sequence of functions as follows:

where

is chosen so that

for continuity. Then if we make

by piecing together the functions

and their domains, this function satisfies all your given conditions but is not differentiable at any integer value of

, where the gradient changes abruptly.

Last edited by scientia (2013-01-16 11:11:58)

Fistfiz

Member

Registered: 2012-07-20

Posts: 33

Re: Little derivative problem (prove or give a counterexample)

uhmm i'm trying to figure out how do you see that f goes to a y0... let's see:

so the c(n) part converges and the x part of course does. Is there a easiest way to see that?

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30+2=28 (Mom's identity)

Fistfiz

Member

Registered: 2012-07-20

Posts: 33

Re: Little derivative problem (prove or give a counterexample)

However, we observe that the derivative (where it does exist) goes to zero... maybe moving (1) to the hypothesis... Anyway, that was a good example, thank you

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30+2=28 (Mom's identity)