Question 24410: Missing Components, Missing Partials, Nondifferentiability and Saddles Need Different Layouts

Original question and givens

Use mode=nondifferentiable_norm, positive scale=10, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partials_exist, gradient_defined, boundary).

• G001 — Use mode=nondifferentiable_norm, positive scale=10, with the source-declared explicit counterexample for that mode. Give (mode, function_description, point, coordinate_partials_exist, gradient_defined, boundary).

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-gradient-vector

Step-by-step derivation

source roles = [gradient-boundary-counterexample]

derived = [[mode, function_description, point, coordinate_partials_exist, gradient_defined, boundary]]

source roles = [nondifferentiable_norm, 10]

derived = [nondifferentiable_norm, f(x,y)=10*sqrt(x^2+y^2), [0, 0], No, No, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined]

Reverse checks

• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.

Result

(nondifferentiable_norm, f(x,y)=10*sqrt(x^2+y^2), [0, 0], No, No, coordinate_partials_fail_at_the_origin_so_no_gradient_is_defined)

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.