Answer
a) Not continuous
b) Continuous
c) Continuous
d) Continuous
e) Not continuous
f) Continuous
Work Step by Step
Continuity: for all $x$-values in the interval, the limit at that value exists and is equal to the function output at that value.
For the endpoints of closed intervals, the half-limit from the side within the interval (right-hand limit for lower bounds, left-hand limit for upper bounds) is used instead of the full limit.
a) $f(3)$ does not have a value, therefore the limit at that point is not equal to the function's value at the point, so the function is not continuous on this interval.
b) All points in (1,2) and (2,3) form continuous lines. Despite not being smooth at $x=2$, the function is continuous there too: $\lim\limits_{x \to 2^{-}} =\lim\limits_{x \to 2^{+}} = 0$, and therefore $\lim\limits_{x \to 2} = 0 = f(2)$
c) All points in (1,2) form a continuous interval, and both endpoints of the closed interval are filled circles attached to the line, therefore their half-limits are equal to their actual values.
d) All points in (1,2) form a continuous interval
e) See explanation for (a)
f) All points in (2,3) form a continuous interval
