My advice, from the point of ease-of-use (and also because that's the only thing visible in your post) is to use \section
instead of \part
with the following added to your preamble:
\renewcommand{\thesection}{\Roman{section}}
Here is a minimal example highlighting this output:

\documentclass{article}
\usepackage{amssymb,lipsum}
\renewcommand{\thesection}{\Roman{section}}
\begin{document}
\section{Functions from $\mathbb{R}^{n}$ to $\mathbb{R}^{m}$}
\lipsum
\end{document}
If you must use \part
, here's a modification (patch, via etoolbox
) to the \part
-related macro that sets it the way you want:

\documentclass{article}
\usepackage{amssymb,etoolbox,lipsum}
\renewcommand{\thesection}{\Roman{section}}
\makeatletter
\patchcmd{\@part}% <cmd>
{\Large\bfseries\partname\nobreakspace\thepart\par\nobreak}% <search>
{\huge\bfseries\thepart\quad}% <replace>
{}{}% <success><failure>
\makeatother
\begin{document}
\part{Functions from $\mathbb{R}^{n}$ to $\mathbb{R}^{m}$}
\lipsum
\end{document}