Umattinog768

2023-03-18

How to find the continuity of a function on a closed interval?

srnessgebf

Beginner2023-03-19Added 8 answers

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

Another thing we need to do is to

Demonstrate that a function on a closed interval is continuous.

This is determined by the specific 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, Thus, 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 piecemeal (by cases) must be examined using the above considerations, paying special attention to the numbers at which the rules change.

Other types of functions are addressed in the class or through the definition.

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

Another thing we need to do is to

Demonstrate that a function on a closed interval is continuous.

This is determined by the specific 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, Thus, 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 piecemeal (by cases) must be examined using the above considerations, paying special attention to the numbers at which the rules change.

Other types of functions are addressed in the class or through the definition.

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