What is the Definition of Continuity for Functions and How Can It Be Proven?

[curlyc3]

Hi! I'm struggling with analysis at the moment and here's one question that I'm struggling with! COuld somebody please explain what I need to do, using a delta/epsilon definition
The question is:

Define what it means for a function f to be continuous at a point a.

(a) Prove directly from your definition that f(x) = x^3 is continuous
everywhere.

(b) Give an example of a function that is continuous except at
x = 1, where it is not continuous.

Thanks

 

[heman]

u can see the reply of Matt Grime in the post epsilon and delta...

 

[mathman]

(b) Give an example of a function that is continuous except at
x = 1, where it is not continuous.

f(x)=0 ,x<1
f(x)=1 ,x>1

 

[arildno]

(b) Give an example of a function that is continuous except at
x = 1, where it is not continuous.

Thanks

$$f(x)=2.3456712345, x\neq{1}, f(1)=-\frac{\pi^{tan(Sinh(0.5))}}{3.77}$$