À differentiable function f: (a, b) → R is uniformly differentiable on a, b if for every E>0 there exists a d > 0 such that | f(t) – f(x) く€ t - I or all t, r E [a, b] with 0< t- x| < 6. Show that f is uniformly differentiable on [a, b] E and only if f' is continuous on a, b.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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À differentiable function f: a, b) → R is uniformly differentiable on a, b if for every
E>0 there exists a o> 0 such that
|10) – {(e) – r) <«
f(t)- f(x)
t - x
for all t, r E [a, b] with 0< t- a| < 8. Show that f is uniformly differentiable on [a, b]
if and only if f' is continuous on [a, b).
Transcribed Image Text:À differentiable function f: a, b) → R is uniformly differentiable on a, b if for every E>0 there exists a o> 0 such that |10) – {(e) – r) <« f(t)- f(x) t - x for all t, r E [a, b] with 0< t- a| < 8. Show that f is uniformly differentiable on [a, b] if and only if f' is continuous on [a, b).
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