You can use the Darboux sums (named after the mathematician Gaston Darboux). If you have a totally ordered metric space $M$ with a minimum and a maximum ($\alpha$ and $\omega$) (this is, in particular, the case of a curve with an injective parametrization, but you do not have to refer to it). Call subdivision a sequence of points $s=(x_i)_{0\leq i\leq n}$ linearly ordered $(\forall i<n)(x_i<x_{i+1})$ and joining the endpoints $x_0=\alpha,\ x_n=\omega$. For every subdivision $s$, form the Darboux sum \begin{equation} l(s)=\sum_{i=1}^n d(x_i,x_{i-1})\ . \end{equation} Take the usual order of refinement between subdivisions, if these quantities converge for the net of refinement order, the limit the length of the linearly ordered metric space.

$$ length(M)=lim_{s\nearrow} l(s)\ . $$

You can use the Darboux sums (named after the mathematician Gaston Darboux). If you have a totally ordered metric space $M$ with a minimum and a maximum ($\alpha$ and $\omega$) (this is, in particular, the case of a curve with an injective parametrization, but you do not have to refer to it). Call subdivision a sequence of points $s=(x_i)_{0\leq i\leq n}$ linearly ordered $(\forall i<n)(x_i<x_{i+1})$ and joining the endpoints $x_0=\alpha,\ x_n=\omega$. For every subdivision $s$, form the Darboux sum \begin{equation} l(s)=\sum_{i=1}^n d(x_i,x_{i-1})\ . \end{equation} Take the usual order of refinement between subdivisions, if these quantities converge for the net of refinement order, the limit the length of the linearly ordered metric space.

$$ length(M)=lim_{s\nearrow} l(s)\ . $$

Call rectifiable a (linearly ordered) metric space such that $length(M)<+\infty$. This notion is elementary, encompasses all others I know and has very nice properties :

• if a (linearly ordered) metric space is rectifiable then all its intervals $[u,v]$ are so
• length is additive : if $u<v<w$, then $$ length([u,w])=length([u,v])+length([v,w]) $$

You can use the Darboux sums (named after the mathematician Gaston Darboux). If you have a totally ordered metric space with a minimum and a maximum ($\alpha$ and $\omega$) (this is, in particular, the case of a curve with an injective parametrization, but you do not have to refer to it). Call subdivision a sequence of points $s=(x_i)_{0\leq i\leq n}$ linearly ordered $x_i<x_{i+1}$ and joining the endpoints $x_0=\alpha,\ x_n=\omega$. Take the usual order of refinement and form the Darboux sum \begin{equation} l(s)=\sum_{i=1}^n d(x_i,x_{i-1}) \end{equation} if these quantities converge for the net of refinement, call it the length of the linearly ordered metric space.

You can use the Darboux sums (named after the mathematician Gaston Darboux). If you have a totally ordered metric space $M$ with a minimum and a maximum ($\alpha$ and $\omega$) (this is, in particular, the case of a curve with an injective parametrization, but you do not have to refer to it). Call subdivision a sequence of points $s=(x_i)_{0\leq i\leq n}$ linearly ordered $(\forall i<n)(x_i<x_{i+1})$ and joining the endpoints $x_0=\alpha,\ x_n=\omega$. For every subdivision $s$, form the Darboux sum \begin{equation} l(s)=\sum_{i=1}^n d(x_i,x_{i-1})\ . \end{equation} Take the usual order of refinement between subdivisions, if these quantities converge for the net of refinement order, the limit the length of the linearly ordered metric space.

$$ length(M)=lim_{s\nearrow} l(s)\ . $$