Function in Differentiability Class 1 is also in Continuity Class
Jump to navigation
Jump to search
Theorem
Let $f$ be a real function.
Let $f$ be an element of differentiability class $C^1$.
Then $f$ is also an element of the class $C$ of continuous real functions.
Proof
By definition of $C^1$, $f \in C^1$ if and only if $f$ is differentiable.
By definition of $C$, $f \in C$ if and only if $f$ is continuous.
The result follows from Differentiable Function is Continuous.
$\blacksquare$
Sources
- 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 2$. Functions of One Variable: $2.2$ Derivatives