Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ • π ƒ -¹ ² ³ °
You are not logged in.
#1 2013-01-17 05:49:29
- Fistfiz
- Member
Offline
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.
30+2=28 (Mom's identity)
#2 2013-01-17 06:22:12
- scientia
- Full Member
Offline
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 (2013-01-17 10:11:58)
