Solving Limits and Continuity Problems with Examples and Proofs

[vbplaya]

I'm having a little trouble trying to figure out these problems. Any help would be appreciated.

g(x) = (x^2 - a^2)/(x-a) when x≠a but 8 when x=a... how do i find the constant a so that the function will be continuous on the entire real line?


f(x)= x^3 - x^2 + x - 2 on closed interval [0,3] f(c)=4. How do I find the value of c that is guaranteed by the Intermediate Value Theorem?
---I've proven via IVT that there exists a 0 in [0,3] but I do not know how to find the c value.


if f(x)=g(x) for x≠c and f(c)≠g(c) then either f or g is not continuous at c. True or False.
--- I haven't a clue. I can't even think of an example where f(x)=g(x) but f(c)≠g(c).


this last one I just want to make sure I'm doing it right.
Show that the Dirichlet function f(x)= 0 if x is rational and 1 if x is irrational
is not continuous at any real number.

if I just write D(x) = lim m→∞ lim n→∞ cos^2n (m! pi x) is that showing that the function is not continuous at any real number?

Again, I'd appreciate any help or pointers in the right direction. Thanks.

 

[stunner5000pt]

cant u simplify the first one such that you get g(x) = x+a and a cna be any real number?

ALso for the continuity proof
use the DEFINITION OF CONTINUITY that is
$$\lim_{x \rightarrow c} f(x) = f(c)$$ if $$\forall \epsilon >0, \exists \delta>0$$ such that if $$0<|x-c|< \delta$$ then $$|f(x)-f(c)| <\epsilon$$

you have to use the definition to prove continuity. Who says that that formula u wrote is continuous or not?