Understanding Interval Continuity

Instructor: Matthew Bergstresser

Matthew has a Master of Arts degree in Physics Education. He has taught high school chemistry and physics for 14 years.

Continuity means contiguous or continuing. Graphs of functions are either continuous between two x-values boundaries or not. In this lesson, we will investigate how to determine if a function is continuous in an interval.

Wet Concrete

Imagine walking down the sidewalk between fences on either side of the sidewalk and there is a section of the side walk that is roped off. Concrete was just poured along the section and is still wet so you can't go any further. The continuity of the sidewalk is broken over the interval that has wet concrete. This is an analogy for graphs of functions that are not continuous over an interval. An interval is the portion of a function between two x-values. Let's look deeper into interval continuity.

Interval Continuity

A function is continuous along an interval if each value along the interval is valid. The formal definition is

Let's say we are interested in interval (a,c]. The parenthesis in front of the ''a'' indicates a is not part of the interval and the square bracket after the ''c'' indicates c is included in the interval. The interval (2, 5] includes x = 5, but not x = 2. Let's look at some functions and intervals to determine whether the graph is continuous along the interval.

Function 1

Let's determine whether the function is continuous along the interval (-5,0].

Looking at the graph, we can start to the left of point p estimating where -5 is and trace the curve with our finger towards zero. Our finger stops at point p because we can see the graph stops. To get to the remaining part of the function we need to pick up our finger to go to the origin (0,0). This means the function is not continuous along the interval (-5,0] because the x values between point p and 0 are not defined. Let's look at another example using Function 1.

Is Function 1 continuous over the interval (-5,p)? We can trace the function just like we did last time, but this time it is continuous because the point p is not included in the interval. Let's leave Function 1 and look at a new function.

Function 2

Is Function 2 continuous in the interval (-p,q]?

Tracing the curve from -p we can get to point q without lifting our finger, which means the function is continuous along the interval (-p,q]. What about along the interval (q,r)? Tracing the path along the curve from point q and ending just before reaching point r shows the function is continuous along the interval. The reason we stopped before point r is because it is not included in the interval. Let's look at one more function and practice determining whether the function exhibits interval continuity.

Function 3

Determine if Function 3 is continuous on the interval (-p,0).

The graph is continuous between these points, but since these points are not included in the interval the function is not continuous on this interval.

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