Answer
A,B,D, and E
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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A has an open interval as the domain, no breaks (discontinuities). Continuous.
B is not defined for x=1, so we observe the two intervals that make up the domain, $(-\infty,1)\cup(1,+\infty)$. It is continuous on each of them, therefore it is continuous on its domain
C At x=1 there is a discontinuity, because f(1)=2 and $\displaystyle \lim_{\mathrm{x}\rightarrow 1}f(x)$=1.
D the answer is the same as for case B
E The domain is $(-\infty,-1]\cup(1,+\infty) . $The left interval has a right endpoint, and the left limit equals 1, as does f(-1). f is continuous on the left interval.
It is also continuous on the right interval.
Therefore, it is continuous on its domain
F. The domain has a left endpoint at x=1.
f(1)=2, but the right limit of f(x) there is equal to 1.
so, at x=1, there is a discontinuity
