The idea is: No matter how close (##\delta##) you get to ##x=a##, there is always a positive distance (##\varepsilon##) such that there are image points (##f(a),f(x)##) at least so far from another.Miguel said:Hey Guys, I posed this on Math Stackexchange but no one is offering a good answering. I though you guys might be able to help :)
https://math.stackexchange.com/questions/3049661/single-point-continuity-spivak-ch-6-q5
Single point continuity, also known as pointwise continuity, is a property of a function where the limit of the function exists at a specific point in its domain. This means that the function is continuous at a single point, but may not necessarily be continuous at other points.
Uniform continuity is a stronger form of continuity that requires the function to be continuous at every point in its domain. In contrast, single point continuity only requires the function to be continuous at a specific point in its domain.
Yes, it is possible for a function to be uniformly continuous but not single point continuous. This can occur when the function has a discontinuity at a single point, but the discontinuity is "small enough" that it does not affect the overall continuity of the function.
The epsilon-delta definition of continuity is a mathematical way of describing the concept of single point continuity. It states that a function is continuous at a point if, for any small positive number (epsilon), there exists a small positive number (delta) such that the difference between the function's output at any point within delta distance of the input point is less than epsilon.
Yes, a function can have multiple points of single point continuity. This means that the function is continuous at each of these points, but may not be continuous at other points in its domain.