y=x^3 (continuity)

May 2010
1
0
use the definition of epsilon-delta to proof f(x)=x^3 is continuous
pls help to solve this
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
use the definition of epsilon-delta to proof f(x)=x^3 is continuous
pls help to solve this
To show that \(\displaystyle f(x)= x^3\) is continuous at x= a, you must show that
"Given any \(\displaystyle \epsilon> 0\), there exist \(\displaystyle \delta> 0\) such that if \(\displaystyle |x- a|< \delta\) then \(\displaystyle |x^3- a^3|< \epsilon\)."

It will help to recognize that \(\displaystyle |x^3- a^3|= |x- a||x^2+ ax+ a^2|\) so you will need to find an upper bound on \(\displaystyle |x^2+ ax+ a^2|\).
 
Mar 2009
419
64
Uptown Manhattan, NY, USA
If you're still stuck, observe that if \(\displaystyle |x-a| < 1\), then \(\displaystyle |x| < 1 + |a|\) and then follow from that.