I think I am making this harder

• Jun 3rd 2006, 10:16 AM
Susie38
I think I am making this harder
the definition of continutiy is stated in garbled forms. Identify the one that is equivalent for the definition of continuity and draw a sketch of what each of the others represents.

a. for every epsilon > 0 and every delta > 0, |x - c| < delta implies |f(x) - f(c)| < epsilon.

b. There is an epsilon >0 such that for every delta>0, |x - c| <delta implies
|f(x) - f(c)| < epsilon.

c. for some epsilon > 0, there is a delta > 0 such that |x - c| < delta implies |f(x) - f(c)| < epsilon.

d. There is a delta >0 such that for every epsilon >0, |x - c| <delta implies
|f(x) - f(c)| < epsilon.

I chose d. I don't know for sure. :confused:
• Jun 3rd 2006, 06:53 PM
ThePerfectHacker
Quote:

Originally Posted by Susie38
the definition of continutiy is stated in garbled forms. Identify the one that is equivalent for the definition of continuity and draw a sketch of what each of the others represents.

a. for every epsilon > 0 and every delta > 0, |x - c| < delta implies |f(x) - f(c)| < epsilon.

b. There is an epsilon >0 such that for every delta>0, |x - c| <delta implies
|f(x) - f(c)| < epsilon.

c. for some epsilon > 0, there is a delta > 0 such that |x - c| < delta implies |f(x) - f(c)| < epsilon.

d. There is a delta >0 such that for every epsilon >0, |x - c| <delta implies
|f(x) - f(c)| < epsilon.

I chose d. I don't know for sure. :confused:

THe definition for coutinouity of real function is that,
$\displaystyle \lim_{x\to c}f(x)=f(c)$
By definition we have,
$\displaystyle \forall \epsilon>0$ there is a $\displaystyle \delta>0$ such as, $\displaystyle |f(x)-f(c)|<\epsilon$ whenever $\displaystyle |x-c|<\delta$.

The meaning of "whenever" is the same as this implies something else. Thus it is a saying,
.... $\displaystyle |x-c|<\delta$ implies $\displaystyle |f(x)-f(c)|<\epsilon$ thus (d) is the correct choice.
• Jun 4th 2006, 08:28 AM
Nichelle14
How do I use a graph to represent the other 3 choices that were not the right answer?