Any 'normal' continuous functioncan be expressed as a polynomial. A 'normal' function has a smooth curve with no corners, so that we can differentiate it as many times as we need to with no restrictions. To say that the function is continuous on an interval means that the graph of the function may be drawn without lifting pen from the paper. The graph below is continuous but not differential. It cannot be represented by a Maclaurin series because it is not differentiable on any interval containing 0.
The polynomial that represents f(x) may be infinite – that is, it may have an infinite number of terms - and if the function is not one to one, so that each value on the range corresponds to only one value in the domain, then the domain may have to be restricted to an interval on which the function is one to one. Given this we may expressas a sum of powers ofmultiplied by differentials of
In this expressionmeanshas been differentiatedtimes thensubstituted.
The are several points to notes.
Puttingeliminates all but the first term on the right hand side and the left hand side is
Differentiating both sides and substitutinggiveson the left hand side. The first term on the right hand side disappears, the second term becomesand all succeeding terms disappear becauseso both sides equalif both sides are differentiated. In general if we differentiate both sidestimes then putthen both sides equal
Ifis a polynomial of degreethen all but the firstterms vanish. Ifthen: