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## Homework Statement

A function

*f*has a simple zero (or zero of multiplicity 1) at x

_{0}if

*f*is differentiable in a neighborhood of x

_{0}and

*f*(x

_{0}) = 0 while

*f*(x

_{0}) ≠ 0.

Prove that

*f*has a simple zero at x

_{0}iff

*f*(x) =

*g*(x)(x - x

_{0}), where

*g*is continuous at x

_{0}and differentiable in a deleted neighborhood of x

_{0}, and

*g*(x

_{0}) ≠ 0.

## Homework Equations

None.

## The Attempt at a Solution

You need to find a function g(x) such that it fufills the properties in the problem. I thought of using the function g(x) = f(x)/(x-x

_{0}), but that is not continuous at x

_{0}. Am I close in trying to find this function g or should the proof go in a different direction?