Prove that for each X E (9 b)
lim f'x, = lim f'x) X = f'x, (x)
and lim f(x) = lim fix) = fox
x 1 x.
(iv) Let x be a point in (ab). Prove that
(fis differentiable at X
) (fis continuous at x K (fiscontinuous. at x )
(vi) Early in the course last term, we showed that for each real number B > 0
and integer pal, there exists a unique positive real number x such that XP = B.
We then used that result to show that for each real number B> there exists
a unique monotone increasing function g: R (0,00) such that g(1)=B =
and, for all X, YETR
glxty) = georgly).
Why does it follow at once that g is everywhere differentiable,
and that, for all XE TR,
(We usually write B*
gix, = gengged?
for glx, of course. )
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