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If $f$ and $g$ are continuous at $x=0$ and $g(0)\neq 0$, which of the following is necessarily continuous at $x=0$?
A$\dfrac{f(x)}{g(x)}$
B$\dfrac{1}{f(x)}$ always
C$f(x)$ only if $g(x)$ is differentiable
D$f\circ g$ only if $f$ is discontinuous
Answer & Solution
Correct answer: A. $\dfrac{f(x)}{g(x)}$
By the algebra of continuous functions, sums, products, and quotients are continuous wherever defined. Since $g(0)\neq 0$, $\dfrac{f(x)}{g(x)}$ is defined near $x=0$ and is continuous there. Option B is not always true because it requires $f(0)\neq 0$.
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