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Bacle2
#3
Feb17-12, 05:02 PM
Sci Advisor
P: 1,170
Quote Quote by Flying_Goat View Post
A function defined on ℝ is continuous at x if given ε, there is a δ such that |f(x)-f(y)|<ε whenever |x-y|<δ. Does this imply that f(x+δ)-f(x)=ε? The definition only deals with open intervals so i am not sure about this. If this is not true could someone please show me a counter example for it?

Any help would be appreciated. Thanks.
Another approach is :

The description can also be interpreted as saying that one can find, for any ε>0, a value of δ>0 every point x in the interval:

(y-δ,y+δ) on the x-axis

Is mapped into the interval (f(y)-ε,f(y)+ε )

on the y-axis.

Try playing with relatively-simple functions like x2, and see what happens with
the expression f(x+δ)-f(x), for different values of δ, and how you can choose δ to make the difference be within ε.