Calculus- Early Transcendentals, 2021a

3.7. Continuity 107
Definition 3.45: Continuous from the Right and from the Left
A function f is left continuous at a point a if
and right continuous at a point a if
lim f (x)= f (a)
x→a− lim f (x)= f (a).
x→a +
If a function f is continuous at a, then it is both left and right continuous at a.
The above definition regarding left (or right) continuous functions is illustrated with the following
figure:
One-sided limits allows us to extend the definition of continuity to closed intervals. The following
definition means a function is continuous on a closed interval if it is continuous in the interior of the
interval and possesses the appropriate one-sided continuity at the endpoints of the interval.
Definition 3.46: Continuity on a Closed Interval
A function f is continuous on the closed interval [a,b] if:
i. it is continuous on the open interval (a,b);
ii. it is left continuous at point a:
and
iii. it is right continuous at point b:
lim f (x)= f (a);
x→a− lim f (x)= f (b).
x→b +