why is a piecewise function f(x) left differentiable at c if and only if f(left)=f(c)? rule for differentiability
2026-03-25 16:14:05.1774455245
why is a piecewise function f(x) left differentiable at c if and only if f(left)=f(c)?
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It's not. You may have missed seeing the hypothesis (just above this claim on the linked page) that $f_1$ and $f_2$ are differentiable everywhere.
For instance, the function $f(x) = \begin{cases}x & \text{$x$ rational and negative} \\0 & \text{$x$ irrational and negative} \\ 0 & \text{$x$ positive} \end{cases}$
has a left limit that equals $f(0)$, but is certainly not left-differentiable.