<p>This paper investigates a path-following method inspired by the semismooth^∗ approach for solving algebraic inclusions, with a primary emphasis on the role of strong subregularity and its crucial role in ensuring the robustness and stability of the path-following method, as it provides a framework to uniformly control the distance between the input and the solution set across a continuous path. We explore the problem of finding a mapping <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <mi>x</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">]</mo> <mo>⟶</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>n</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$x: [0, T] \longrightarrow \mathbb{R}^{n} $</EquationSource> </InlineEquation> that satisfies <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <mn>0</mn> <mo>∈</mo> <mi>F</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>,</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$0 \in F(t, x(t)) $</EquationSource> </InlineEquation> for each <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mi>t</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">]</mo> </math></EquationSource> <EquationSource Format="TEX">$t \in [0, T] $</EquationSource> </InlineEquation>, where <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <mi>F</mi> </math></EquationSource> <EquationSource Format="TEX">$F $</EquationSource> </InlineEquation> is a set-valued mapping from <InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">R</mi> <mo>×</mo> <msup> <mi mathvariant="double-struck">R</mi> <mi>n</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$\mathbb{R} \times \mathbb{R}^{n} $</EquationSource> </InlineEquation> to <InlineEquation ID="IEq6"> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="double-struck">R</mi> <mi>n</mi> </msup> </math></EquationSource> <EquationSource Format="TEX">$\mathbb{R}^{n} $</EquationSource> </InlineEquation>. We consider an approach based on mappings exhibiting uniform semismooth^∗ properties along continuous trajectories, thereby maintaining a uniform grid error throughout the interval. Error estimates demonstrate an <InlineEquation ID="IEq7"> <EquationSource Format="MATHML"><math> <mi>o</mi> <mo stretchy="false">(</mo> <mi>h</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$o(h)$</EquationSource> </InlineEquation> grid error with respect to the discretization parameter under natural regularity assumptions. Numerical experiments applied to electric circuit and elastoplastic models confirm the efficiency and robustness of the method.</p>