Function Continuity and Properties

Function Continuity

• A function f(x) is said to be continuous at a point a if the following three conditions are satisfied:
  1. f(a) is defined
  2. lim x→a f(x) exists
  3. lim x→a f(x) = f(a)
• In other words, a function is continuous at a point if the limit of the function as x approaches a is equal to the value of the function at a.

Identifying Continuity of a Function

• A function can be identified as continuous if it satisfies the following properties:
  • The function is defined at the point: The function has a value at the point in question.
  • The function has a limit at the point: The limit of the function as x approaches the point exists.
  • The limit equals the function value: The limit of the function as x approaches the point is equal to the value of the function at that point.
• Methods to identify continuity include:
  • Graphical analysis: Visual inspection of the graph to check for gaps, holes, or jumps.
  • Limit analysis: Calculating the limit of the function as x approaches the point to check for existence and equality.
  • Algebraic analysis: Simplifying the function and checking for discontinuities.

Properties of Continuity

• Sum Rule: The sum of two continuous functions is continuous.
• Product Rule: The product of two continuous functions is continuous.
• Chain Rule: The composition of two continuous functions is continuous.
• Inverse Rule: The inverse of a continuous function is continuous, if it exists.
• Intermediate Value Theorem: If a function is continuous on a closed interval and takes on values f(a) and f(b) at the endpoints, then it takes on all values between f(a) and f(b) at some point in the interval.
• Extreme Value Theorem: A continuous function on a closed interval takes on a maximum and minimum value on that interval.