= Improper line integral
{title2=$\lim_{\varepsilon\downarrow0}\int_{a+\varepsilon}^b\mathbf F(\mathbf r(t))\cdot\mathbf r'(t)\,dt$}
A <line integral> can be defined as an <improper integral> of its parametrized integrand when that integrand is undefined or unbounded at isolated parameter values. At an interior singular parameter, split the integral and require both one-sided limits to exist separately; their cancellation in a symmetric <Cauchy principal value> is a different convention. For a scalar <arc length> integral, use $f(\mathbf r(t))|\mathbf r'(t)|$ in place of the vector-field <dot product>.
If the restricted integrand has a removable singularity, integrating its continuous extension gives the same value. This does not require the original <vector field> to extend continuously to a full neighbourhood of the curve. Likewise, a differential identity may hold only on the regular segments: a zero integrand there has improper integral zero, although its printed classical expression may not be defined at the omitted points.
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