Differentiability Class/Examples/Class Zero Function

Example of Differentiability Class

Let $f$ be the real function defined as:

$\map f x = \begin {cases} 0 & : x < 0 \\ x & : x \ge 0 \end {cases}$

Then $f \in C^0$ but $f \notin C^1$.

Proof

By inspection it is seen that $f$ is continuous everywhere.

We have that:

$\map {f'} x = \begin {cases} 0 & : x < 0 \\ 1 & : x \ge 0 \end {cases}$

Hence $f'$ is not continuous at $x = 0$.

So by definition of differentiability class, $f$ is not a member of $C^1$.

$\blacksquare$

Sources

• 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 2$. Functions of One Variable: $2.2$ Derivatives: Example $\text F$