Which of the following statements are true? I. If f and g are continuous at a, so is f + g. II. If f and g are continuous at a, so is f/g, whatever g(a) is. III. If f is continuous at a, so is | f |. IV. If | f | is continuous at a, so is f.

Question 66

MediumOriginal question

Which of the following statements are true?
I. If $f$ and $g$ are continuous at $a$ , so is $f + g$ .
II. If $f$ and $g$ are continuous at $a$ , so is $\frac{f}{g}$ , whatever $g(a)$ is.
III. If $f$ is continuous at $a$ , so is $\lvert f \rvert$ .
IV. If $\lvert f \rvert$ is continuous at $a$ , so is $f$ .

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

Show the answer and explanation

CorrectNot quiteAnswer D

Correct answer: D

I and III are true. II needs $g(a) \ne 0$ .
IV is false: $f(x) = 1$ for $x \ge 0$ , $-1$ for $x < 0$ has $\lvert f \rvert = 1$ , continuous, while $f$ jumps at 0.

Worth noting

Taking the modulus can hide a jump, so IV runs the wrong way.

Concept: The algebra of continuous functions
Typical timeTypical: 120 s
Your timeYours: —

The rule this testsContinuous functions

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