Prove that any derivative of a given function is bounded Let the function $f\left( x \right) = \left( {\frac{{1 - \cos x}}
{{{x^2}}}} \right)\cos (3x)$ if $x\ne 0$ and $f(0)=\frac{1}{2}$. Prove that any derivative of $f$ is bounded on $\mathbb{R}$. Thank so much for helping.

The original function is an entire function hence its derivatives are bounded on $[-1,1]$. Moreover, any derivative of $\frac{1-\cos x}{x^2}$ is bounded on $\mathbb{R}\setminus[-1,1]$ since it is a linear combination of functions of the form $\frac{f(x)}{x^k}$ where $f(x)$ is bounded trigonometric function and $k\geq 2$; so the same holds for $\frac{1-\cos x}{x^2}\cos(3x)$ over $\mathbb{R}$.
