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Making a piecewise function continuous

Problem 1.762 · medium

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

Answer \( k = \frac{2}{3} \)

Lines: 3 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Reviewed	reviewers: gpt-oss:20b, qwen3.6:27b-mlx	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: pass
qwen3.6:27b-mlx: pass — The solution correctly identifies the condition for continuity at the piecewise boundary and solves for k accurately.

Every verdict on record (6)

qwen3.6:27b-mlx: pass 2026-10-25 — The solution correctly identifies the condition for continuity at the piecewise boundary and solves for k accurately.
gpt-oss:20b: pass 2026-10-25
qwen3.6:27b-mlx: pass 2026-10-24 — The solution correctly identifies the condition for continuity at the boundary and solves for k accurately.
gpt-oss:20b: pass 2026-10-24
qwen3.6:27b-mlx: fail (error) 2026-10-24 — The solution fails to explicitly state the continuity condition (equating the left and right limits) before solving for k. Line 4 jumps directly to substituting the answer, omitting the crucial step of setting 3k - 4 = -2.
gpt-oss:20b: pass 2026-10-24

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 2026-10-25 with SymPy 1.14.0.