# parameter

• Apr 12th 2011, 04:58 PM
surjective
parameter
Hello,

I have the following function:

$\begin{displaymath}f(x) = \left\{\begin{array}{lr}\frac{1}{\sqrt{x}} & : x \in [1,\infty[\\0 & : x \notin [1,\infty[\end{array}\right.\end{displaymath}$

How do I calculate the exact value of the parameter $p \in [1,\infty[$ so that $f \in L^{p}(\mathbb{R})$?
• Apr 12th 2011, 11:52 PM
Moo
Hello,

Well just compute $\displaystyle\int |f(x)|^p ~dx=\int_1^\infty \frac{1}{x^{1/2+p}} ~dx$

This is a Riemann-type integral and it converges iff $\frac 12+p>1$