Making a piecewise function continuous
Problem 1.208 · medium
Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k x + 2 & x < 3 \\ 2 x & x \ge 3 \end{cases} \) continuous for all \( \displaystyle x \).
- Each piece is a polynomial, so f is continuous everywhere except possibly at x = 3. There the left-hand limit, the right-hand limit and f(c) must agree.
- \[ \lim_{x \to 3^-}\left(k x + 2\right) = 3 k + 2 \]The left-hand limit, in terms of k.✓ Proved
- \[ \lim_{x \to 3^+}\left(2 x\right) = 6 \]The right-hand limit, which is also f(c).✓ Proved
- \[ 6 \]Setting them equal gives k = 4/3.✓ Proved
Answer \( k = \frac{4}{3} \)
Lines: 3 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. A reviewer's objection to the explanation stands; see the receipt.
The full receipt
| Line | Status | Checked by | Detail |
|---|---|---|---|
| 1 | Not checked | — | a sentence; read, not computed |
| 2 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 3 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| 4 | ✓ Proved | sympy 1.14.0 | simplify(lhs - rhs) reduced to 0 |
| answer, a second way | ✓ Checked independently | sympy 1.14.0 + mpmath 1.3.0 | with that k the two pieces agree at c ± 1e-10 |
Reviewers
gpt-oss:20b: fail (error) — Sentence 4 incorrectly states "6 = 6" instead of equating the left‑hand limit 3k+2 to the right‑hand limit 6 and solving for k. This misstates the conclusion and would mislead a student.qwen3.6:27b-mlx: fail (error) — The solution fails to actually solve for k. It states the condition for continuity (limits must agree) but the final equation shown is '6 = 6', which is a tautology and does not involve k. The step should have been '3k + 2 = 6' leading to 'k = 4/3'.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to actually solve for k. It states the condition for continuity (limits must agree) but the final equation shown is '6 = 6', which is a tautology and does not involve k. The step should have been '3k + 2 = 6' leading to 'k = 4/3'.gpt-oss:20b: fail (error) 2026-10-04 — Sentence 4 incorrectly states "6 = 6" instead of equating the left‑hand limit 3k+2 to the right‑hand limit 6 and solving for k. This misstates the conclusion and would mislead a student.qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution fails to actually solve for k. It states the condition for continuity (limits must agree) but then presents the tautology '6 = 6' instead of the equation '3k + 2 = 6'. The variable k is completely absent from the final step, making the derivation of the answer impossible to follow.gpt-oss:20b: fail (error) 2026-10-04 — The solution incorrectly equates the limits as 6 = 6 instead of setting 3k + 2 = 6 and solving for k. This misstates the necessary condition for continuity.
Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate
method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence,
not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by
generator:structured/continuity_parameter, checked 2026-10-04 with SymPy 1.14.0.