What are the three criteria for a function to be continuous?

Answer 1

the three conditions for a function f(x) to be continuous at x=a
1.f(a) must exist
2.limit x tending to a must exist
3.limit x tending to a must be equal to f(a)

Answer 2

To state that a function f(x) is continuous at x = a, it must be true that:
1. The function is defined at x = a
2. The limit of the function as x approaches to a exists
3. The limit of the function at x = a is equal to f(a) ( the value of the function at this point)

Answer 3

Actually, for f to be continuous at a point a of its domain, it's not necessary that the limit of f exist at a and equal f(a). This is indeed necessary and sufficient if a is an accumulation point of the domain of f.

According to the so-called epsilon- delta definition, we say a function f is continuous at a point a of it's domain D if, for every eps>0, there exists delta >0 (depending on eps), such that if x is in D and |x - a| < delta then |f(x) - f(a)| < eps. This is equivalent to saying that, for every sequence (a_n) in D which converges to a, the image sequence (f(a_n)) converges to f(a).

Answer 4

Left hand limit= limit at a point= Right hand Limit then it is said to be continuous at that point

Answer 5

The nth derivates must be continuous.