Can anyone help on this problem:

Supposef is defined on (−∞,∞) and |f(x) − f(y)| ≤ K|x− y|^α

for all x, y E(−∞,∞) and some constants K and α> 0 .

Show that f is continuous on (−∞,∞) .

If α> 1 , show that f '(x) = 0 for all x E(−∞,∞)

Thank you.

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- Jul 21st 2008, 12:14 PMGreen03[SOLVED] Urgent
Can anyone help on this problem:

Supposef is defined on (−∞,∞) and |f(x) − f(y)| ≤ K|x− y|^α

for all x, y E(−∞,∞) and some constants K and α> 0 .

Show that f is continuous on (−∞,∞) .

If α> 1 , show that f '(x) = 0 for all x E(−∞,∞)

Thank you.

- Jul 21st 2008, 12:35 PMPlato
If $\displaystyle \varepsilon > 0$ define $\displaystyle \delta = \left( {\frac{\varepsilon }{K}} \right)^\frac {1}{a} $.

- Jul 21st 2008, 01:27 PMGreen03
is that only to show is continuous?

- Jul 21st 2008, 01:46 PMPlato
$\displaystyle \left| {f(x) - f(y)} \right| \le K\left| {x - y} \right|^a \Rightarrow \quad \left| {\frac{{f(x) - f(y)}}{{x - y}} - 0} \right| \le K\left| {x - y} \right|^{a - 1} $