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 £© Section
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 £© Continuous Functions
Given the work of the previous section, we are now in a position to state a clear definition
of the notion of continuity. We will have several related definitions, but the fundamental
definition is that of continuity at a point. Intuitively, continuity at a point c for a function
f means that the values of f for points near c do not change abruptly from the value of f
at c. Section has shown that, mathematically, this means that as x approaches c, the
value of f(x) must be approaching f(c). Hence we have the following basic definition.
Definition We say that a function f is continuous at a point c if
lim f(x) = f(c). ()
x→c
It is important to note that this definition places three conditions on the behavior of
the function f near the point c. Namely, f is continuous at the point c if (1) f is defined
at c, (2) lim f(x) exists, and (3) lim f(x) = f(c).
x→c x→c
Corresponding to one-sided limits, we have the notions of continuity from the left and
from the right.
Definition We say that a function f is continuous from the left at a point c if
lim f(x) = f(c). ()
x→c−
We say that a function f is continuous from the right at a point c if
lim f(x) = f(c). ()
x→c+
Simply to say that a function f is continuous, without specifyi