The statement is given ''A polynomial function is continuous for all real numbers" .
Consider the polynomial
[tex]f(x)=a_0+a_1x+_{}a_1x^2\ldots\ldots\ldots\ldots\ldots\ldots\ldots\text{.}\mathrm{}a_nx^n_{}[/tex]Since every polynomial function is valid for every rela number.
Prove continuity for the polynomial function at any point c.
[tex]\lim _{x\rightarrow c}f(x)=f(c)[/tex]For LHS,
[tex]\lim _{x\rightarrow c}f(x)=\lim _{x\rightarrow c}(a_0+a_1x+\ldots\ldots\ldots\ldots a_nx^n)[/tex]Susbtitute x=c.
[tex]a_0+a_1c_{}+\ldots\ldots\ldots\ldots\ldots\ldots\ldots.a_nc^n[/tex]For RHS
[tex]f(c)=a_0+a_1c+\ldots\ldots\ldots\ldots\ldots\ldots..a_nc^n[/tex]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.
