Making a piecewise function continuous
Problem 1.554 · medium
Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k x + 2 & x < 1 \\ 1 - 3 x & x \ge 1 \end{cases} \) 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
- \[ \lim_{x \to 1^+}\left(1 - 3 x\right) = -2 \]The right-hand limit, which is also f(c).✓ Proved
- \[ -2 \]Setting them equal gives k = -4.✓ Proved
Answer \( k = -4 \)
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 two limits as -2 = -2 instead of setting k+2 = -2, which would yield k = -4.qwen3.6:27b-mlx: fail (error) — The solution sets the left-hand limit (k + 2) equal to the right-hand limit (-2) but incorrectly writes the equation as -2 = -2. This omits the variable k entirely and fails to show the actual equation k + 2 = -2 required to solve for k.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution sets the left-hand limit (k + 2) equal to the right-hand limit (-2) but incorrectly writes the equation as -2 = -2. This omits the variable k entirely and fails to show the actual equation k + 2 = -2 required to solve for k.gpt-oss:20b: fail (error) 2026-10-11 — Step 4 incorrectly equates the two limits as -2 = -2 instead of setting k+2 = -2, which would yield k = -4.gpt-oss:20b: fail (misleading) 2026-10-11 — The solution incorrectly states the equality as "-2 = -2" instead of "k+2 = -2", which could mislead a student about how to solve for k.qwen3.6:27b-mlx: fail (error) 2026-10-11 — The solution claims that setting the limits equal results in the equation '-2 = -2', which is a tautology and does not solve for k. The correct equation should be 'k + 2 = -2'.
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 2026-10-11 with SymPy 1.14.0.