Which of these functions are continuous at x = 1? I. f(x) = (x^2 + x - 2)/(x - 1) for x ≠ 1, and f(1) = 3 II. f(x) = x - | x - 1 | III. f(x) = [x] + | x - 1 | IV. f(x) = 2x for x < 1, and f(x) = x^2 + 1 for x ≥ 1

Question 23

MediumOriginal question

Which of these functions are continuous at $x = 1$ ?
I. $f(x) = \frac{x^{2} + x - 2}{x - 1}$ for $x \ne 1$ , and $f(1) = 3$
II. $f(x) = x - \lvert x - 1 \rvert$
III. $f(x) = [x] + \lvert x - 1 \rvert$
IV. $f(x) = 2x$ for $x < 1$ , and $f(x) = x^{2} + 1$ for $x \ge 1$

I, II and IV onlyCorrect answerYour answerI and IV onlyCorrect answerYour answerAll four: I, II, III and IVCorrect answerYour answerII and IV onlyCorrect answerYour answer

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CorrectNot quiteAnswer A

Correct answer: A

I: $\frac{x^{2} + x - 2}{x - 1} = x + 2 \to 3 = f(1)$ . II: a difference of continuous functions. IV: both pieces give 2 at $x = 1$ . All three are continuous.
III: just left of 1, $[x] + \lvert x - 1 \rvert \to 0$ , while at 1 it is $1 + 0 = 1$ . A jump.

Worth noting

The corner of $\lvert x - 1 \rvert$ in II does not break continuity; the jump of $[x]$ in III does.

Concept: Testing continuity at a point
Typical timeTypical: 120 s
Your timeYours: —

The rule this testsContinuity at a point

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