Continuity of Polynomial Functions in their Domain

In summary, the conversation discusses the continuity of polynomial functions on a closed interval and the use of concise mathematical notation to express this concept. It is noted that all polynomials are defined, continuous, and differentiable on the entire real line.
  • #1
Miike012
1,009
0
The first hypothesis is that f is continuous on [a,b]...

Is there a more concise mathematical way of saying... "because the function f is a polynomial it is continuous in its domain."? Because I rather not write that on my test it looks sloppy and non professional...
 
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  • #2
I don't see any problem with what you said. All polynomials are defined and continuous on the entire real line.
 
  • #3
I know but is there another way to state that using math symbols or something to make it more concise in like 2-5 words.
 
  • #4
This is all you need to say.
"...because the function f is a polynomial, it is defined and continuous [STRIKE]in its domain[/STRIKE] on the entire real line."
 
  • #5
Assuming you are using standard notation: "If p is a polynomial, then p ε C[a,b]".
 
  • #6
Note, by the way, that Rolle's theorem also requires that the function be differentiable on some interval. Fortunately, it is also true that all polynomials are differentiable for all x.
 

1. What is the definition of continuity for polynomial functions?

Continuity for polynomial functions means that the function is uninterrupted and has no gaps or breaks in its graph. This means that the function is defined and can be evaluated at every point in its domain.

2. How can I determine if a polynomial function is continuous at a specific point?

To determine if a polynomial function is continuous at a specific point, evaluate the function at that point. If the function is defined and has a finite value at that point, then the function is continuous at that point.

3. Can a polynomial function be continuous at one point but not at another?

Yes, a polynomial function can be continuous at one point and not at another. This is because continuity at a point depends on the value of the function at that specific point, so it is possible for the function to have breaks or gaps at other points.

4. Are all polynomial functions continuous?

No, not all polynomial functions are continuous. Some polynomial functions may have breaks, gaps, or asymptotes in their graphs, which means they are not continuous in their entire domain.

5. How can I determine if a polynomial function is continuous in its entire domain?

To determine if a polynomial function is continuous in its entire domain, you can check for any breaks, gaps, or asymptotes in its graph. If there are none, then the function is continuous in its entire domain. Additionally, you can use the rules for continuity, such as the sum, difference, and product rules, to determine if a function is continuous in its domain.

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