Which of the following statements are true?
I. If $f$ is continuous at $a$ and $g$ is not, then $f + g$ is not continuous at $a$ .
II. If neither $f$ nor $g$ is continuous at $a$ , then $f + g$ is not continuous at $a$ .
III. If $f$ is continuous at $a$ and $g$ is not, then $fg$ is not continuous at $a$ .
IV. If $f$ is continuous at $a$ and $f(a) \ne 0$ , then $\frac{1}{f}$ is continuous at $a$ .

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

Show the answer and explanation

CorrectNot quiteAnswer B

Correct answer: B

I is true: if $f + g$ were continuous at $a$ , so would be $(f + g) - f = g$ . IV is true: a quotient of continuous functions is continuous where the denominator is not 0.
II is false: $[x]$ and $-[x]$ both jump at 0, but their sum is 0. III is false: $x \operatorname{sgn} x = \lvert x \rvert$ is continuous at 0, though $\operatorname{sgn} x$ is not.

Worth noting

Two jumps can cancel in a sum, and a factor that is 0 at $a$ can cancel a jump in a product.

Concept: The algebra of continuous functions
Typical timeTypical: 150 s
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The rule this testsContinuous functions

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