Continuity of a function !

meer

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Mar 18, 2017
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Good Day!
the above is from munem calculus, chapter 2 , set 2.11 and topic is "change of sign property"
It says "polynomial function f is continuous on the interval [0, 1]"
my question is how do we know from above that function is continuous on the stated interval ?
Thank you.
 
The way this is written is a bit confusing. You need to be aware that it is a general theorem about polynomials

Every polynomial is continuous and infinitely differentiable in
(, )(- \infty, \ \infty)
.

So steps in the reasoning based on this general theorem have been left implicit

All polynomials are continuous in (,)    The polynomial f(x) is continuous in any interval    f(x) is continuous in [0,1].\text {All polynomials are continuous in } (-\infty, \infty) \implies\\ \text {The polynomial } f(x) \text { is continuous in any interval} \implies \\ f(x) \text { is continuous in } [0, 1].
It would be clearer English to write

As a polynomial, f(x) is continuous in [0, 1]
 
I agree with the others.
You can't just say that f(x) is continuous on [0,1].
It has been next to impossible to get my students to explain why f(x) is continuous on an interval.
As JeffM stated, f(x) is a polynomial and all polynomials are continuous on any interval. Hence f(x) is continuous on [0,1]
 
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