Learning Objectives
• Given a function, determine if it is continuous at a certain point using the three criteria for continuity.
• Evaluate a one-sided limit.
• Determine if a function is continuous on a closed interval using one-sided limits.
• Apply the above objectives to the greatest integer function.
Continuity: The Big Picture
• Continuous functions are the “normal” functions. They are the ones that do not have any holes, asymptotes, jumps, or matter that break the smooth flow of the function.
• The functions on the right are continuous throughout
y = x
y = x2
Continuity at a Point
• The functions on the right are not continuous at x = 0, but they are continuous at all of the other points.
• You can usually tell by looking if a function is continuous at a certain point.
• However, in calculus, we have a more formal definition of continuity.
• The functions on the right are y = 1/x and y = |x|/x.
• For both functions, f(0) does not exist, for it is undefined.
• Therefore, the functions are discontinuous at x = 0
Pre-Example 1
• Observe the function on the right. Let’s see if it is continuous at x = 1.
• It certainly meets the first criteria, for f(1) exists. In fact, f(1) = 1.
• However, the limit as x1 does not exist.
• Therefore, the function is not continuous at x = 1
Pre-Example 2
• Observe the function on the right. Is it continuous at x = 1?
• First criteria: f(1) = 3. f(1) exists!
• The limit as x1 is 1. The limit exists too!
• However, f(1) and the limit do not equal. Fails third criteria.
Example 1
Determine if the following function is continuous at h(1) and h(2).
• (Use your three criteria.)
One-sided Limits
We’ve learned:
The limit as x approaches c from the left.
The limit as x approaches c from the right.
)(lim xfcx
The overall limit. This exists only if the left and right limits are equal.
The Greatest Integer Function
• This is also known as the step function. You can probably see why in the graph of y = ||x|| on the right.
• To evaluate the above function for a given x, determine the integer that is equal to or just below x.
• The greatest integer function is one in which one-sided limits are especially used.
• For example, what is:
xx 2lim
xx 2lim
Interval Review
Remember:
• (a, b) is called an open interval. It is the range of every single number from a to b, not including a and b.
• [a, b] is called a closed interval. It is like an open interval except that it includes a and b.
Continuity on an Open Interval
• A function is continuous on the open interval (a, b) if it is continuous at every point in that interval.
Continuity on a Closed Interval
A function is continuous on the closed interval [a, b] if:
• It is continuous on the open interval (a, b)
• The following one-sided limits exist:
Example 3
Determine if the function below is continuous on the interval [-1, 1].
Remember:
• First determine if it is continuous on (-1, 1) (You can do this by graphing.)
• Then evaluate the one-sided limits at -1 and 1.
Wrap-Up
• Know how to determine if a function is continuous at a point using the three criteria.
• Know one-sided limits.
• Know the greatest integer function.
• Know how to determine continuity on intervals.