Making a piecewise function continuous
Problem 1.300 · medium
Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k x - 2 & x < 1 \\ - 2 x & x \ge 1 \end{cases} \) continuous for all \( \displaystyle x \).
- Each piece is a polynomial, so f is continuous everywhere except possibly at x = 1. There the left-hand limit, the right-hand limit and f(c) must agree.
- \[ \lim_{x \to 1^-}\left(k x - 2\right) = k - 2 \]The left-hand limit, in terms of k.✓ Proved
- \[ \lim_{x \to 1^+}\left(- 2 x\right) = -2 \]The right-hand limit, which is also f(c).✓ Proved
- \[ -2 \]Setting them equal gives k = 0.✓ Proved
Answer \( k = 0 \)
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) — Step 4 incorrectly equates the limits; it should set k-2 = -2, which yields k = 0. The solution as written would mislead a student into thinking the equality holds without solving for k.qwen3.6:27b-mlx: fail (error) — Step 4 claims that setting the limits equal yields k = 0, but the equation should be k - 2 = -2, which implies k = 2. The solution incorrectly solves the resulting equation.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-05 — Step 4 claims that setting the limits equal yields k = 0, but the equation should be k - 2 = -2, which implies k = 2. The solution incorrectly solves the resulting equation.gpt-oss:20b: fail (error) 2026-10-05 — Step 4 incorrectly equates the limits; it should set k-2 = -2, which yields k = 0. The solution as written would mislead a student into thinking the equality holds without solving for k.qwen3.6:27b-mlx: fail (error) 2026-10-05 — Step 4 claims that setting the limits equal (-2 = -2) yields k = 0, but the equation actually derived from equating the limits is k - 2 = -2, which implies k = 2. The solution contains a logical contradiction and an incorrect final answer.gpt-oss:20b: fail (error) 2026-10-05 — The solution incorrectly sets the left‑hand limit equal to the right‑hand limit. It writes "-2 = -2" instead of "k-2 = -2", so the conclusion that k = 0 is not justified by the equations presented.
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 2026-10-05 with SymPy 1.14.0.