1.12 Confirming Continuity over an Interval

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Simple polynomials are continuous everywhere; it's the trigonometric, rational, and piecewise functions that might have continuity problems.

The concept of continuity over an interval is quite simple; if the graph of the function doesn’t have any breaks, holes, or other discontinuities within a certain interval, the function is continuous over that interval. However, this definition of continuity changes depending on your interval and whether the interval is closed or open. 

Confirming continuity over an open interval is fairly easy:

To put it simply, if you can trace the function from a to b without picking up your pencil, then the function is continuous over the interval. However, it is important to remember that this is an open interval. So, you actually start with your pencil at the point directly after a and ending at the point directly before b. ✏️

Confirming continuity over a closed interval is a bit more complicated. 

Confirming continuity over a closed interval requires both the function being continuous at every point within the interval, as well as the limit of the function as x approaches the endpoint of the interval being equal to the value of the function at that endpoint. This is known as the "two-sided limit test" for continuity over a closed interval.

To put it in a formal way, for a function f(x) to be continuous over a closed interval [a, b], the following must be true:

It's important to note that the limits test for the left and the right sides of the interval, this is why it's called "two-sided limit test"

Practice Problems:

4. Confirm that k(x) = sin(x) is continuous over the closed interval [0, pi]

5.Confirm that m(x) = 1/(x-2) is continuous over the open interval (-1, 3)

6.Confirm that n(x) = sqrt(x+3) is continuous over the closed interval [-3, 4]

7. Confirm that p(x) = 3x^2 + 2x - 1 is continuous over the open interval (-2, 2)

8. Confirm that q(x) = e^x is continuous over the closed interval [-1, 1]

9. Confirm that r(x) = ln(x+2) is continuous over the open interval (0, 3)

10. Confirm that s(x) = 1/x^2 is continuous over the closed interval [-3,3]

Solving these practice problems will give you a better understanding of the concepts of continuity over an open interval and a closed interval. It's important to remember the difference between the two, and to use the correct definition of continuity when working with different types of intervals.

In conclusion, confirming continuity over an interval is an important concept in calculus, as it helps us understand the behavior of a function and its smoothness. It's a simple concept to understand, but it requires a bit of practice to be able to apply it to different types of functions and intervals. Remembering the definition of continuity over an open interval and a closed interval is crucial, and it should be the starting point of solving any problem related to continuity over an interval.