Every continuous singular simplex is smooth

The path $\sigma (t)=\mathrm{\mid }t-1\mathrm{/}2\mathrm{\mid }$ on $[0,1]={\mathrm{\Delta }}^1$ is continuous, since $\mathrm{\mid }\mathrm{\mid }s-1\mathrm{/}2\mathrm{\mid }-\mathrm{\mid }t-1\mathrm{/}2\mathrm{\mid }\mathrm{\mid }\le \mathrm{\mid }s-t\mathrm{\mid }$. At its interior point $1\mathrm{/}2$ the right difference quotients are one and the left difference quotients are minus one. Therefore it is not differentiable there.

Any extension in [F1] would restrict to a differentiable function on an interval around $1\mathrm{/}2$ agreeing with $\sigma$. Its derivative would have to equal both limits from step 1.1, an impossibility. Thus this is a continuous singular one-simplex which is not smooth.

Its endpoints both equal $1\mathrm{/}2$, but it is nonconstant, so endpoint agreement does not fix the interior defect. All zero-simplices and constant simplices in this target are smooth by constant extension; the empty target has no witness. The explicit formula uses no choice and requires no boundary-target convention.