Differentiability Class/Examples/Class 1 Function
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Example of Differentiability Class
Let $f$ be the real function defined as:
- $\map f x = \begin {cases} 0 & : x < 0 \\ x^2 & : x \ge 0 \end {cases}$
Then $f \in C^1$ but $f \notin C^2$.
Proof
By inspection it is seen that $f$ is continuous everywhere.
We have that:
- $\map {f'} x = \begin {cases} 0 & : x < 0 \\ 2 x & : x \ge 0 \end {cases}$
By inspection it is seen that $f'$ is continuous everywhere.
So by definition of differentiability class, $f$ is a member of $C^1$.
Then we have that:
- $\map {f''} x = \begin {cases} 0 & : x < 0 \\ 2 & : x \ge 0 \end {cases}$
Hence $f''$ is not continuous at $x = 0$.
So by definition of differentiability class, $f$ is not a member of $C^2$.
$\blacksquare$
Sources
- 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 2$. Functions of One Variable: $2.2$ Derivatives: Example $\text F$