Derivative iff Right and Left Derivative
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Theorem
Let $f$ be a real function.
Then $f$ is differentiable if and only if it has both right- and left-hand derivatives which agree.
Proof
Derivatives are defined in terms of limits.
The result follows from Limit iff Limits from Left and Right.
$\blacksquare$
Sources
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): left and right derivatives