## 3.2 Limits and Continuity

(This topic appears in Section 3.2 in Applied Calculus or Section 10.2 in Finite Mathematics and Applied Calculus)

Let us look once again at the graph we examined in the previous tutorial:

Notice that the graph has two "breaks" in it: one at x = 0, and the other when x = 1. We refer to such breaks as discontinuities.
Continuous Functions

The function f(x) is continuous at x = a if

 limx→a f(x) exists That is, the left-and right limits exist and agree with each other limx→a f(x) = f(a)

The function f is said to be continuous on its domain if it is continuous at each point in its domain. If f is not continuous at a particular point a, we say that f is discontinuous at a or that f has a discontinuity at a.

Example

Refer to the graph we have been studying:

Q Is f continuous at x = -2?
A Check the definition:
 lim x→-2 f(x) exists, and equals 2.
 f(-2) also equals 2.
Therefore, f(x) is continuous at x = -2.

Q Is f continuous at x = 0?
A Again check the definition:
 limx→0 f(x) does not exist.

Therefore, f(x) is not continuous at x = 0.

Q Is f continuous at x = 1?
A Referring to the definition:
 limx→1 f(x) = 1
 f(1) = -1

Since the limit at 1 does not agree with f(1), f(x) is not continuous at x = 1.

Back in the tutorial on functions from the graphical viewpoint, we looked at the following function:

f(x)=  -1 if -4 x < -1 x if -1 x 1 whose graph is: x2-1 if 1 < x 2
 Q The function f is is not continuous at x = 0. Q The function f is is not continuous at x = -1. Q The function f is is not continuous at x = 1.

Now look at the following graphs. None of the functions shown is defined at x = 10. However, two of them can be made continuous by defining f(10) = 15. Click on those two graphs.

You will see more about continuous functions in the next tutorial when we discuss them algebraically.

Now try the rest of the exercises in Section 3.2 in Applied Calculus or Section 10.2 in Finite Mathematics and Applied Calculus

Alternatively, go on to the algebraic approach.

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Last Updated: August, 2007