Classifying Differentiability of Absolute Powers

Introduction

For a function involving an absolute value, the expression on the two sides of zero can behave differently. The absolute value itself is nonnegative, but the difference quotient divides by the signed increment \(h\). That sign can determine whether the two one-sided limits agree.

For the family \(f(x)=|x|^p\), the exponent controls how quickly the numerator approaches zero compared with the denominator. We can classify differentiability at zero directly from the definition, without relying on a general differentiation rule.

The Function and Its Difference Quotient

For \(p>0\), define \(f:\mathbb{R}\to\mathbb{R}\) by \(f(x)=|x|^p\), with \(f(0)=0\). The derivative at zero exists as a finite real number exactly when the difference quotient has a finite limit as \(h\) approaches zero:

Here, \(f(0)=0\), so for \(h\ne 0\) the quotient is \(\frac{|h|^p}{h}\). To take its limit, we must retain the sign of \(h\).

When \(h>0\), \(|h|=h\), so the quotient is \(h^{p-1}\). When \(h<0\), \(|h|=-h\), and division by the negative \(h\) gives a minus sign:

These expressions provide the key comparison. For \(p>1\), the positive power \(p-1\) makes both sides approach zero. For \(p=1\), the two sides are different constants. For \(0<p<1\), the magnitude grows without bound, with opposite signs.

Worked Examples

Classification for Positive Exponents

The examples illustrate all three behaviors for \(p>0\). The full classification follows by examining \(p-1\) in the one-sided formulas.

This argument proves the classification directly from the derivative definition. In particular, a quotient that becomes unbounded is not a finite derivative, even if it has a definite direction of growth on each side.

Why the Sign Must Be Kept

A common mistake is to replace the signed denominator \(h\) by \(|h|\) when simplifying. That changes the quotient. For example, an incorrect calculation for \(f(x)=|x|\) might say

The error is that \(h\) and \(|h|\) are not equal when \(h<0\). The correct calculation keeps the denominator unchanged:

The correction matters: the incorrect calculation would predict a derivative of \(1\), but the actual one-sided limits disagree. Whenever the increment can be negative, retain its sign until the one-sided limits have been evaluated.

What About Zero or Negative Exponents?

The classification theorem assumed \(p>0\), so that \(|0|^p=0\) and the function is defined at zero. For \(p<0\), the expression \(|x|^p\) is not defined at \(x=0\). Since a derivative at a point requires the function to be defined at that point, the expression as given has no derivative at zero.

For \(p=0\), the expression \(0^0\) is not defined by the usual real exponentiation convention. Thus \(|x|^0\), without an additional definition at zero, also does not define a function there. If one separately defines the value at zero, the outcome depends on that choice. For example, assigning \(f(0)=1\) while \(f(x)=1\) for \(x\ne0\) gives the constant function, whose derivative at zero is zero. This is a separately defined extension, not a conclusion from evaluating \(0^0\).

More generally, assigning a value at zero changes the difference quotient, so the positive-exponent theorem cannot simply be applied to a different definition. The domain and the value at the point are part of the differentiability question.

Why This Classification Matters

The exponent provides a compact way to compare the size of the numerator with the size of the increment. When \(p>1\), the numerator \(|h|^p\) shrinks faster than \(|h|\), leaving a quotient that approaches zero. At \(p=1\), the numerator and denominator shrink at the same rate, but the sign change across zero produces unequal slopes. When \(0<p<1\), the numerator shrinks more slowly than \(|h|\), so the quotient becomes unbounded in magnitude.

These are distinct reasons for failing to have a finite derivative. Checking both one-sided quotients identifies which behavior occurs and avoids treating every nondifferentiable case as though it were the same.

Summary

For positive exponents, the difference quotient of \(f(x)=|x|^p\) at zero has right-hand form \(h^{p-1}\) and left-hand form \(-(-h)^{p-1}\). If \(p>1\), both limits are zero and the derivative exists. If \(p=1\), the limits are \(1\) and \(-1\). If \(0<p<1\), the quotients are unbounded in magnitude. For \(p\leq0\), the expression does not by itself define a function at zero under the usual convention, so differentiability there requires a separately specified value.