Differentiability and least squares

Here is an interesting problem I encountered in Mike Wilson’s analysis course: let \(u \in \mathbb{R}^d \) be a unit vector and define the function \(f: \mathbb{R}^d \rightarrow \mathbb{R} \) by

Show that \(f\) is differentiable on all of \(\mathbb{R}^d\), and find an expression for \(f’(p)\) in terms of \(p\) and \(u\).

Check it out here!

N.B.: Professor Wilson’s class is quite hard (this is one of the easier problems he assigned in the second half of the semester) but well worth the effort. I highly recommend it to any interested student.

Written on June 30, 2017