List of mathematical formulas: “Continuity, points of discontinuity

List of basic formulas and concepts related to continuity and discontinuity points:

Continuity of function

1. Definition of continuity at a point:
   A function ( f(x) ) is continuous at a point ( a ) if three conditions are met:

• ( f(a) ) is defined.
• ( \lim_{x \to a} f(x) ) exists.
• ( \lim_{x \to a} f(x) = f(a) ).

1. Continuity on an interval:
   A function ( f(x) ) is continuous on an interval ( [a, b] ) if it is continuous at every point of this interval.

Breaking points

1. Definition of a point of discontinuity:
   A point (a) is a point of discontinuity of a function (f(x)), if at least one of the continuity conditions is not satisfied.
2. Types of breaks:

• Discontinuity of the first kind (or discontinuity with a finite limit):
• ( \lim_{x \to a} f(x) ) exists, but ( \lim_{x \to a} f(x) \neq f(a) ).
• Discontinuity of the second kind (or infinite discontinuity):
• ( \lim_{x \to a} f(x) ) does not exist (for example, ( f(x) \to \infty ) or ( f(x) \to -\infty )).

1. Examples:

• The function ( f(x) = \frac{1}{x} ) has a discontinuity of the second kind at the point ( x = 0 ).
• The function ( f(x) = \begin{cases}
  x^2, & x < 1 \ 2, & x = 1 \ x + 1, & x > 1
  \end{cases} ) has a discontinuity of the first kind at the point ( x = 1 ).