Does a bounded function converge if its derivative tends to zero?

First idea: If you drop the boundedness assumption, $\ln(x)$ does provide a counterexample to what you want.

Now think about how you can transform this counterexample into a bounded counterexample.

Hint

Typical bounded functions include $\sin$ and $\cos$.

Tags:

Derivatives

Real Analysis

Related

The degree of $\sqrt{2} + \sqrt[3]{5}$ over $\mathbb Q$ Is there any function which grows 'slower' than its derivative? How to efficiently represent and manipulate polynomials in software? Find the value of : $\lim\limits_{n\to \infty} \sqrt [n]{\frac{(3n)!}{n!(2n+1)!}} $ Show $\operatorname{Aut}(C_2 \times C_2)$ is isomorphic to $S_3=D_6$ Polylogarithms of negative integer order In how many ways can the ball come back to $A$ after seven passes? Show that the arithmetic mean is less or equal than the quadratic mean How to solve this ordinary differential equation? Does there exist a space $W$ s.t. $X\to W,\ Y\to W$ are covering spaces when $Z\to X,\ Z\to Y$ are covering spaces? Prove that vector space and dual space have same dimension If $G$ is a compact topological group, how to show that a finite index subgroup of $G$ is open?

Recent Posts