Continuous Functions

[Jason76]

1. \(\displaystyle y = 3x^{2} - 7\) is free of discontinuities

2. \(\displaystyle y = -3sin(x)\) is continuous over the entire set of real numbers. Why is there a difference between ex 1 and ex 2?

3. \(\displaystyle y = 4csc(x)\) has discontinuities where x = 0 or \(\displaystyle \pi\) I understand this is true because sin = 0 at such points, therefore the inverse of 0 is undefined.

 

[stapel]

1. \(\displaystyle y = 3x^{2} - 7\) is free of discontinuities

2. \(\displaystyle y = -3sin(x)\) is continuous over the entire set of real numbers. Why is there a difference between ex 1 and ex 2?

One is said to be free of discontinuities (which means that it is continuous everywhere, being the entire set of real numbers) and the other is said to be continuous over the entire set of real numbers (which means that it is free of discontinuities). What is the "difference" between them?

3. \(\displaystyle y = 4csc(x)\) has discontinuities where x = 0 or \(\displaystyle \pi\) I understand this is true because sin = 0 at such points, therefore the inverse of 0 is undefined.

What is your question here?

 

[JeffM]

1. \(\displaystyle y = 3x^{2} - 7\) is free of discontinuities

2. \(\displaystyle y = -3sin(x)\) is continuous over the entire set of real numbers. Why is there a difference between ex 1 and ex 2?

3. \(\displaystyle y = 4csc(x)\) has discontinuities where x = 0 or \(\displaystyle \pi\) I understand this is true because sin = 0 at such points, therefore the inverse of 0 is undefined.

Because someone is not being consistent in their language. Such inconsistency is not helpful to the student. You see different wordings and wonder whether a different meaning is intended.

 

[HallsofIvy]

Because someone is not being consistent in their language. Such inconsistency is not helpful to the student. You see different wordings and wonder whether a different meaning is intended.

I remember many years ago trying to learn how to use a Tektronix graphics terminal. They had an example of getting the coordinates of the cursor in which they accessed "x" as an integer and "y" as a string that then was converted to an integer! Confused me terribly.