Function in Differentiability Class 1 is also in Continuity Class

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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.

$\blacksquare$


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