Question

Proving Continuity.
The theorem says
limx->x0 f(x) = f(x0)

I'm confused. Continuity means there is no break in the line at any point or can we only prove for continuity at a point that we choose (x0)? I see x as the entire domain of the function, and x0 as just a point, but this would prove continuity only at x0 right?

Answer 1

Right. Continuity can be proven at a point OR for a complete function. Each has it's purpose.

Eg. \[f(x) =x ^{-1}\] is NOT continuous at x=0 but is continuous for every other value of x.

By continuity being defined for a point we can then use that definition to prove it for one point of a function or all of it as is needed.

PS. x is just a variable and doesn't need to define all number for. For example imagine the function f(x)=x^2 that only exists between -1 and 1. By imposing these limits f(x) does not exist if -1<x<1 and is not continuous unless between -1 and 1.

Answer 2

A common definition:
If the limit exists at a given point and the function is equal to the limit at that point, then the function is considered to be continuous at the point.

Answer 3

Which is the theorem in question.