# Show a function is continuous at every point in Reals

• Oct 20th 2009, 02:56 PM
gixxer998
Show a function is continuous at every point in Reals
Let K be greater than 0 and let f : Real numbers --> Real numbers satisfy the condition |f(x)-f(y)| less than or equal to K|x-y| for all x, y in Reals.
Show that f is continuous at every point in c in Reals.
Thank you.
• Oct 20th 2009, 03:31 PM
Plato
Quote:

Originally Posted by gixxer998
Let K be greater than 0 and let f : Real numbers --> Real numbers satisfy the condition |f(x)-f(y)| less than or equal to K|x-y| for all x, y in Reals.
Show that f is continuous at every point in c in Reals.

$\displaystyle \varepsilon > 0\, \Rightarrow \,\delta = \frac{\varepsilon }{2K}$
• Oct 20th 2009, 04:13 PM
gixxer998
Quote:

Originally Posted by Plato
$\displaystyle \varepsilon > 0\, \Rightarrow \,\delta = \frac{\varepsilon }{2K}$

Thank you. How did you come up with this?
• Oct 20th 2009, 04:45 PM
Krizalid
that's Lipschitz condition, and every Lipschitz function is uniformly continuous, hence continuous.

Plato is telling you to pick that $\displaystyle \delta$ to show the uniform continuity.