Why is multline not splitting my equation?

Question

I'd like a simple multiline environment that automatically breaks lines up to fit within the margins. I haven't decided on the end document style so I would like something that adapts well when layouts change. I don't want to set mandatory breakpoints, so I don't think environments like align can help me. However optional breakpoints are fine. I thought I could achieve this with the multline environment, however the equation is not being split into multiple lines, which I thought was the purpose of the environment. I've tried setting optional break points in the second equation but that did not help ether, both equations spill into the margin.

How can I achieve this? Below is an example.

\documentclass[12pt]{article} 
\usepackage{amsmath, amsthm, amssymb, amsfonts}
\begin{document}
\begin{multline}
    \textnormal{continuously~differentiable}
    \subset\textnormal{Lipschitz~continuous}
    \subset\alpha-\textnormal{H\"older~continuous}
    \subset\textnormal{uniformly~continuous}
    = \textnormal{continuous}
\end{multline} 
\begin{multline}
    \textnormal{Lipschitz~continuous} 
    \allowbreak\subset \textnormal{absolutely~continuous}
    \allowbreak\subset \textnormal{bounded~variation} 
    \allowbreak\subset \textnormal{differentiable~a.e.}
\end{multline}
\end{document}

Answer 1

You wrote,

I don't want to set mandatory breakpoints, so I don't think environments like align can help me. However optional breakpoints are fine. I thought I could achieve this with the multline environment, however the equation is not being split into multiple lines, which I thought was the purpose of the environment.

Not quite. As @HenriMenke has already noted in comment, when using multline you must indicate the desired line break points with \\ markers. In that sense, multline is no different from align.

However, for your two examples, the outcome of using multline environments isn't particularly appealing. If you're ok with using explicit break points, I'd like to suggest that you use a nested equation-aligned combination and set the amount by which successive lines are staggered explicitly.

The code used to generate the preceding screenshot uses multiples of 15mu to generate the offsets; you're of course free to choose a smaller or larger amount.

\documentclass[12pt]{article} 
\usepackage{amsmath} % for 'multline' and 'aligned' environments
\begin{document}
\begin{multline}
    \textnormal{continuously~differentiable}\\
    \subset\textnormal{Lipschitz~continuous}\\
    \subset\textnormal{$\alpha$-H\"older~continuous}\\
    \subset\textnormal{uniformly~continuous}\\
    =\textnormal{continuous} % is '=' correct? should it be '\subset'?
\end{multline} 
\begin{multline}
    \textnormal{Lipschitz~continuous} \\
    \subset \textnormal{absolutely~continuous} \\
    \subset \textnormal{bounded~variation} \\
    \subset \textnormal{differentiable~a.e.}
\end{multline}

\begin{equation}\begin{aligned}[b]
    &\textnormal{continuously~differentiable}\\
    &\mkern15mu\subset\textnormal{Lipschitz~continuous}\\
    &\mkern30mu\subset\textnormal{$\alpha$-H\"older~continuous}\\
    &\mkern45mu\subset\textnormal{uniformly~continuous}\\
    &\mkern60mu=\textnormal{continuous}
\end{aligned}\end{equation} 
\begin{equation}\begin{aligned}[b]
    &\textnormal{Lipschitz~continuous} \\
    &\mkern15mu\subset \textnormal{absolutely~continuous} \\
    &\mkern30mu\subset \textnormal{bounded~variation} \\
    &\mkern45mu\subset \textnormal{differentiable~a.e.}
\end{aligned}\end{equation} 
\end{document}