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Making a piecewise function continuous

Problem 1.295 · medium

Find the value of \( \displaystyle k \) that makes \( \displaystyle f(x) = \begin{cases} k x^{2} - 1 & x < 3 \\ x - 4 & x \ge 3 \end{cases} \) 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.
Reviewed
\[ \lim_{x \to 3^-}\left(k x^{2} - 1\right) = 9 k - 1 \]
The left-hand limit, in terms of k.✓ Proved
\[ \lim_{x \to 3^+}\left(x - 4\right) = -1 \]
The right-hand limit, which is also f(c).✓ Proved
\[ -1 \]
Setting them equal gives k = 0.✓ Proved

Answer \( k = 0 \)

Lines: 3 proved, 1 reviewed. 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	Reviewed	claude-sonnet-5-5	claude-sonnet-5-5: This is a sentence, not an equation. Both pieces are polynomials and so continuous on their open domains, leaving only x = 3 as a possible discontinuity. The three-part continuity condition (left limit, right limit, f(3)) is the standard one.
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: pass
qwen3.6:27b-mlx: fail (error) — Step 4 claims that setting the limits equal gives k = 0, but the equation 9k - 1 = -1 actually yields k = 0 only if 9k = 0, which is correct, but the line shown is '-1 = -1', which is a tautology that does not solve for k. The step fails to show the equation 9k - 1 = -1.

Senior review claude-sonnet-5-5, 2026-10-05: fail — The answer k = 0 is correct, since 9k - 1 = -1 gives k = 0. Line 4 does not follow from the previous lines, and the equation that determines k is never written, so a student would not see where k = 0 comes from. It should read 9k - 1 = -1, then k = 0.

qwen3.6:27b-mlx: uphold — The final line is the trivial '-1 = -1' and never shows the equation 9k - 1 = -1 or its solution, so k = 0 appears without justification; the answer itself is correct, but the key step is missing.

Every verdict on record (5)

qwen3.6:27b-mlx: fail (error) 2026-10-05 — Step 4 claims that setting the limits equal gives k = 0, but the equation 9k - 1 = -1 actually yields k = 0 only if 9k = 0, which is correct, but the line shown is '-1 = -1', which is a tautology that does not solve for k. The step fails to show the equation 9k - 1 = -1.
gpt-oss:20b: pass 2026-10-05
claude-sonnet-5-5: fail (misleading) 2026-10-05 — The answer k = 0 is correct, since 9k - 1 = -1 gives k = 0. Line 4 does not follow from the previous lines, and the equation that determines k is never written, so a student would not see where k = 0 comes from. It should read 9k - 1 = -1, then k = 0.
qwen3.6:27b-mlx: fail (error) 2026-10-05 — The solution sets up the continuity condition but fails to equate the left-hand limit (9k - 1) with the right-hand limit (-1). Instead, it incorrectly states '-1 = -1' and concludes k = 0, which is mathematically invalid reasoning.
gpt-oss:20b: pass 2026-10-05

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.