If (f) is continuous on ([a,b]), differentiable on ((a,b)), then (\exists c \in (a,b)):
[
f'(c) = \fracf(b)-f(a)b-a.
]
A function (f) is continuous at (x=a) if
[
\lim_x \to a f(x) = f(a).
]
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[
Lf'(a) = \lim_h \to 0^- \fracf(a+h)-f(a)h,\quad
Rf'(a) = \lim_h \to 0^+ \fracf(a+h)-f(a)h
]
For differentiability, LHD = RHD.
Example: Show (f(x)=|x|) is not differentiable at (x=0).
LHD = (-1), RHD = (+1).