Limits and Continuity

3.2 Limits and Continuity

Q The function f

continuous at x = 0.

Q The function f

continuous at x = -1.

Q The function f

continuous at x = 1.

(This topic is also 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

lim xa	f(x) exists		That is, the left-and right limits exist and agree with each other
lim xa	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 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 the point 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 the point x = 0?
A Again check the definition:

lim
x

f(x) does not exist.

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

Q Is f continuous at the point x = 1?
A

lim
x

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 for Section 1.2, 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

Now look at the following graphs. None of them are 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

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