Continuity and types of discontinuity
Problem 1.758 · hard
Find every point where \( \displaystyle f(x) = \frac{6 x - 24}{x^{2} - 5 x + 4} \) is discontinuous, and classify each discontinuity as removable, a jump, or infinite.
- A rational function is continuous wherever its denominator is not zero, so only the zeros of the denominator can be discontinuities.
- \[ x^{2} - 5 x + 4 = \left(x - 4\right) \left(x - 1\right) \]Factor the denominator.✓ Proved
- \[ \frac{6 x - 24}{x^{2} - 5 x + 4} = \frac{6}{x - 1} \]The factor (x − 4) cancels.✓ Proved
- \[ \lim_{x \to 4^+}\left(\frac{6 x - 24}{x^{2} - 5 x + 4}\right) = 2 \]At x = 4 the limit exists, but f(4) is undefined: a removable discontinuity.✓ Proved
- \[ \left. 6 \right|_{\substack{ x=1 }} = 6 \]At x = 1 the numerator is not 0 while the denominator is, so |f(x)| grows without bound: an infinite discontinuity.✓ Proved
Answer \( \text{removable at } x = 4;\ \text{infinite at } x = 1 \)
Lines: 4 proved, 1 not checked. 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 | 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 |
| 5 | ✓ 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 | zeros of the denominator found by solve, each probed at ±1e-9: {1: 'infinite', 4: 'removable'} |
Reviewers
gpt-oss:20b: fail (error) — Step 5 incorrectly substitutes the numerator: 6*x-24 at x=1 equals -18, not 6. The conclusion of an infinite discontinuity is correct, but the justification is wrong.qwen3.6:27b-mlx: fail (style) — [domain objection, downgraded to style] Line 5 incorrectly substitutes x=1 into the simplified expression 6/(x-1) to get 6, which is undefined (division by zero). The limit is actually infinite, so the conclusion is correct, but the algebraic step shown is mathematically invalid and misleading.
Every verdict on record (6)
qwen3.6:27b-mlx: fail (style) 2026-10-25 — [domain objection, downgraded to style] Line 5 incorrectly substitutes x=1 into the simplified expression 6/(x-1) to get 6, which is undefined (division by zero). The limit is actually infinite, so the conclusion is correct, but the algebraic step shown is mathematically invalid and misleading.gpt-oss:20b: fail (error) 2026-10-25 — Step 5 incorrectly substitutes the numerator: 6*x-24 at x=1 equals -18, not 6. The conclusion of an infinite discontinuity is correct, but the justification is wrong.qwen3.6:27b-mlx: fail (style) 2026-10-24 — [domain objection, downgraded to style] Step 5 incorrectly substitutes x=1 into the simplified function 6/(x-1) to get 6, which is undefined, not 6. Furthermore, concluding an infinite discontinuity requires showing the limit is infinite, not just that the numerator is non-zero at the root of the denominator.gpt-oss:20b: fail (error) 2026-10-24 — Step 5 incorrectly substitutes the numerator: 6*x-24 at x=1 equals -18, not 6. The conclusion about an infinite discontinuity is correct, but the justification is false.qwen3.6:27b-mlx: fail (style) 2026-10-24 — [domain objection, downgraded to style] Step 5 incorrectly substitutes x=1 into the simplified expression 6/(x-1) and claims the result is 6, which is algebraically false (it is undefined). Furthermore, the reasoning for an infinite discontinuity relies on the limit behavior, not the value of the simplified numerator at the point.gpt-oss:20b: fail (misleading) 2026-10-24 — The last sentence incorrectly states "Subs(6, x, 1) = 6" and does not explain that the limit at x=1 diverges to ±∞; it misleads the reader about the nature of the infinite discontinuity.
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_classify, checked 2026-10-25 with SymPy 1.14.0.