\documentclass{article}
\usepackage[english]{babel}
\usepackage[utf8]{inputenc}

\begin{document}

Correct:
\[ \| f \| \]
Completely ignores resizing:
\[ \| f \| \]
Correct:
\[ \left\| \int f \right\| \]
Gibberish:
\[ \left\|| \int f \right\|| \]

\end{document}
