In: Advanced Math
prove by epsilon-delta definition that f:R->R given by f(x)=x^3 is continuous at x=2
Proof : f : R
R is
continuous at x0 if for all
> 0 there
exists
> 0 for all x
R:
Therefore f(x) = x3 and x0 = 2
Therefore for all > 0 there
exists
> 0 for all x
R > 0
( | x - 2 | < |
x3 - 23 | = | (x-2)(x2 + 2x + 4)|
<
where
is bigger
| (x-2)(x2 + 2x + 4)| =
Therefore f(x) = x3 is continuous at x = 2