Answer
f is continuous on its domain.
Work Step by Step
The function f is said to be continuous on its domain if it is continuous at each point in its domain. A discontinuity can occur at $x=a$ if either
a. $\displaystyle \lim_{\mathrm{x}\rightarrow a}f(x)$ does not exist, or
b. $\displaystyle \lim_{\mathrm{x}\rightarrow a}f(x)$ exists but is not equal to $f(a)$.
At endpoints of the domain (if any), we observe the existence of the left or right limit, as appropriate.
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The domain of f is $(-\infty,0)\cup(0,+\infty).$
x=0 is not a potential problem as it is NOT in the domain of f.
(see the Note just before Example 1).
Searching for discontinuities ON the domain ... there are none.
( no "breaks" or "holes" in the graph).
f is continuous on its domain
