Question

How do you find the continuity of a function on a closed interval?

Answer 1

I'm afraid there is a misunderstanding here. See the explanation section, below.

Explanation:

I think that this question has remained unanswered because of the way it is phrased.

The "continuity of a function on a closed interval" is not something that one "finds".

We can give a Definition of Continuity on a Closed Interval

Function `f` is continuous on open interval `(a.b)` if and only if `f` is continuous at `c` for every `c` in `(a,b)`.

Function `f` is continuous on closed interval `[a.b]` if and only if `f` is continuous on the open interval `(a.b)` and `f` is continuous from the right at `a` and from the left at `b`.
(Continuous on the inside and continuous from the inside at the endpoints.).

Another thing we need to do is to
Show that a function is continuous on a closed interval.

How to do this depends on the particular function.
Polynomial, exponential, and sine and cosine functions are continuous at every real number, so they are continuous on every closed interval.
Sums, differences and products of continuous functions are continuous.
Rational functions, even root functions, trigonometric functions other than sine and cosine, and logarithmic functions are continuous on their domains, So, if the closed interval in question contains no numbers outside the domain of rational function `f`, then `f` is continuous on the interval.

Functions defined piecewise (by cases) must be examined using the considerations above, taking particular note of the numbers at which the rules change.

Other types of functions are covered in class or handled using the definition.