Answer
a) Not continuous
b) Not continuous
c) Not continuous
d) Continuous
e) 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) At $x=2$, there is a discontinuity. $\lim\limits_{x \to 2^{-}}f(x) \ne \lim\limits_{x \to 2^{+}}f(x)$, so the limit at that point does not exist.
b) At $x=2$, there is a discontinuity. $\lim\limits_{x \to 2^{-}}f(x) \ne \lim\limits_{x \to 2^{+}}f(x)$, so the limit at that point does not exist.
c) At $x=2$, the left-hand limit (empty circle) is not equal to the actual value of f(2) (the filled circle), so the function is not continuous at this boundary.
d) All points in (1,2) form a continuous interval.
e) All points in (2,3) 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.
f) All points in (2,3) form a continuous interval.
