A function bounded and differentiable, but have an unbounded derivative?

Click For Summary

Homework Help Overview

The discussion revolves around the properties of functions, specifically whether a function can be both bounded and differentiable while having an unbounded derivative. Participants are exploring examples and theoretical constructs related to this concept.

Discussion Character

  • Exploratory, Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants are considering various types of functions and their derivatives, discussing the possibility of bounded functions with unbounded derivatives. Suggestions include using periodic-like functions with decreasing distances between peaks and exploring simple monomials.

Discussion Status

The discussion is ongoing, with participants offering hints and ideas rather than complete solutions. There is an emphasis on exploring different function types and their behaviors, though no consensus has been reached on specific examples.

Contextual Notes

Participants are reminded to provide hints rather than full solutions, aligning with the forum's guidelines for homework help.

danielkyulee
Messages
5
Reaction score
0
Can a function f: (a,b) in R be bounded and diffferentiablle, but have an unbounded derivative. I believe it can, but can not think of any examples where this is true. Anyone have any ideas?
 
Physics news on Phys.org
Sure, so long as you properly choose your domain.

So let's think about this, we want a function whose derivative approaches infinity, but such that the function itself is bounded. Probably the easiest way to do this is to find a function whose derivative goes to zero, then rotate about the axis y=x.

You don't need anything complicated. A monomial will do fine.
 
Try to think of a function with a graph that looks a bit like the graph of a periodic function, but is such that the distance between the peaks get shorter and shorter instead of staying the same.
 
Fredrik said:
Try to think of a function with a graph that looks a bit like the graph of a periodic function, but is such that the distance between the peaks get shorter and shorter instead of staying the same.

Very nice. Perhaps somewhat more complicated, but nice.
 
Sqrt(x), 0 < x < 1.
 
Ray, here in the homework forum it's important to answer with hints, not complete solutions.
 

Similar threads

Bounded functions with unbounded integrals
  • · Replies 3 ·
Replies
3
Views
3K
Calculate this Integral around the Circular Path using Green's Theorem
  • · Replies 3 ·
Replies
3
Views
2K
Examples of functions and sequences
  • · Replies 3 ·
Replies
3
Views
2K
Interpret Riemann sum to determine integral
  • · Replies 2 ·
Replies
2
Views
2K
Solution for differential equation
  • · Replies 2 ·
Replies
2
Views
1K
Discontinuous partial derivatives example
  • · Replies 2 ·
Replies
2
Views
2K
Correct Usage of Partial Derivative Symbols in PDEs
  • · Replies 10 ·
Replies
10
Views
2K
Undergrad How to see that cos(z) is unbounded, but cos(xy)+isin(xy) is bounded?
  • · Replies 8 ·
Replies
8
Views
2K
Show function series involving arctan is not differentiable at x=0
  • · Replies 26 ·
Replies
26
Views
3K
Unbounded Seq.: 3 is an Upper Bound
  • · Replies 16 ·
Replies
16
Views
3K