Continuity and types of discontinuity
Problem 1.399 · hard
Find every point where \( \displaystyle f(x) = \frac{6 - 6 x}{x^{2} + 2 x - 3} \) 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.
- \[ x^{2} + 2 x - 3 = \left(x - 1\right) \left(x + 3\right) \]Factor the denominator.✓ Proved
- \[ \frac{6 - 6 x}{x^{2} + 2 x - 3} = - \frac{6}{x + 3} \]The factor (x − 1) cancels.✓ Proved
- \[ \lim_{x \to 1^+}\left(\frac{6 - 6 x}{x^{2} + 2 x - 3}\right) = - \frac{3}{2} \]At x = 1 the limit exists, but f(1) is undefined: a removable discontinuity.✓ Proved
- \[ \left. -6 \right|_{\substack{ x=-3 }} = -6 \]At x = -3 the numerator is not 0 while the denominator is, so |f(x)| grows without bound: an infinite discontinuity.✓ Proved
Answer \( \text{removable at } x = 1;\ \text{infinite at } x = -3 \)
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.
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: {-3: 'infinite', 1: 'removable'} |
Reviewers
gpt-oss:20b: passqwen3.6:27b-mlx: fail (error) — Step 5 contains a factual error: it claims the numerator is -6 at x = -3, but the numerator 6 - 6(-3) is actually 24. While the conclusion of an infinite discontinuity is correct, the reasoning provided is mathematically false.
Every verdict on record (4)
gpt-oss:20b: pass 2026-10-07qwen3.6:27b-mlx: fail (error) 2026-10-07 — Step 5 contains a factual error: it claims the numerator is -6 at x = -3, but the numerator 6 - 6(-3) is actually 24. While the conclusion of an infinite discontinuity is correct, the reasoning provided is mathematically false.qwen3.6:27b-mlx: fail (error) 2026-10-07 — Step 5 incorrectly substitutes x = -3 into the simplified numerator -6, claiming the numerator is -6. The original numerator 6 - 6x evaluates to 24 at x = -3. While the conclusion (infinite discontinuity) is correct, the reasoning is mathematically false and misleading.gpt-oss:20b: fail (error) 2026-10-07 — The solution incorrectly evaluates the numerator at x = –3 (it writes –6 instead of 24) and therefore misstates the value of the function there. The conclusion that the discontinuity at –3 is infinite is correct, but the reasoning is flawed.
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 2026-10-07 with SymPy 1.14.0.