Making a piecewise function continuous
Problem 1.203 · medium
Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k x + 5 & x < 1 \\ 2 - 2 x & x \ge 1 \end{cases} \) 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.
- \[ \lim_{x \to 1^-}\left(k x + 5\right) = k + 5 \]The left-hand limit, in terms of k.✓ Proved
- \[ \lim_{x \to 1^+}\left(2 - 2 x\right) = 0 \]The right-hand limit, which is also f(c).✓ Proved
- \[ 0 \]Setting them equal gives k = -5.✓ Proved
Answer \( k = -5 \)
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) — The solution incorrectly sets the left‐hand limit equal to the right‐hand limit. It writes "0 = 0" instead of the correct equation k + 5 = 0, which leads to the correct value k = –5 but the reasoning is flawed.qwen3.6:27b-mlx: fail (error) — Line 4 claims that setting the limits equal results in '0 = 0', which is algebraically incorrect and does not solve for k. The correct equation is k + 5 = 0.
Every verdict on record (4)
qwen3.6:27b-mlx: fail (error) 2026-10-04 — Line 4 claims that setting the limits equal results in '0 = 0', which is algebraically incorrect and does not solve for k. The correct equation is k + 5 = 0.gpt-oss:20b: fail (error) 2026-10-04 — The solution incorrectly sets the left‐hand limit equal to the right‐hand limit. It writes "0 = 0" instead of the correct equation k + 5 = 0, which leads to the correct value k = –5 but the reasoning is flawed.qwen3.6:27b-mlx: fail (error) 2026-10-04 — The solution claims that setting the limits equal results in '0 = 0', which is mathematically incorrect. The correct equation is k + 5 = 0, which yields k = -5. The stated equation does not follow from the previous steps.gpt-oss:20b: fail (misleading) 2026-10-04 — The conclusion step incorrectly states 0 = 0 instead of equating the two limits k + 5 = 0 to solve for k = –5.
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.