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The statement, 'A polynomial function is continuous for all real numbers" isA true for all polynomial functions.B. true for some polynomial functions.Oc.C. never true for polynomial functions.

User Michael Coker
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1 Answer

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The statement is given ''A polynomial function is continuous for all real numbers" .

Consider the polynomial


f(x)=a_0+a_1x+_{}a_1x^2\ldots\ldots\ldots\ldots\ldots\ldots\ldots\text{.}\mathrm{}a_nx^n_{}

Since every polynomial function is valid for every rela number.

Prove continuity for the polynomial function at any point c.


\lim _(x\rightarrow c)f(x)=f(c)

For LHS,


\lim _(x\rightarrow c)f(x)=\lim _(x\rightarrow c)(a_0+a_1x+\ldots\ldots\ldots\ldots a_nx^n)

Susbtitute x=c.


a_0+a_1c_{}+\ldots\ldots\ldots\ldots\ldots\ldots\ldots.a_nc^n

For RHS


f(c)=a_0+a_1c+\ldots\ldots\ldots\ldots\ldots\ldots..a_nc^n

Then LHS=RHS.

The function is continuous at x=c.

Hence every polynomial function is continuous for all real numbers.

The correct option is A.

User Rsalmei
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