Continuous Function Example

If

then function f(x) is continuous at a.

A continuous function implies that:

• f(a) is defined in the domain of f(x).
• The limit of f(x) exists at point a.
• The limit of f(x) at point a equals the value of function f(x) at point a.
  
       

In this segment, some theorems are introduced to justify whether a function is continuous or not.

• A function f(x) is continuous from the left at point a, if
  
       
• A function f(x) is continuous from the right at point a, if
  
       
• Assume that f(x) and g(x) are continuous at point a, then:
  • The constant function h(x) = c is continuous for all x at every a.
  • cf(x) is continuous at a for any constant c.
  • f(x) + g(x) is continuous at a.
  • f(x) - g(x) is continuous at a.
  • f(x)g(x) is continuous at a.
  • f(x)/g(x) is continuous at a if g(a) ≠ 0.
  • f^1/n is continuous at a if f(a)^1/n exists.
    
• Every polynomial function
  
       
  
  is continuous.
  
• Every rational function
  
       
  
  is continuous where
  • f(x) and g(x) are polynomial functions.
  • g(x) ≠ 0
    
• If f(x) is continuous at b and
  
       
  
  then
  
       
• If g(x) is continuous at a and f(x) is continuous at g(a), then
  
       
  
  is continuous at a. In other words, a continuous function of a continuous function is a continuous function.

In order to justify whether a value will fall in a specific range or not, intermediate value theorem is introduced.

Assume:

• f(x) is continuous on the interval [a, b]

• f(a) ≠ f(b)
• N is any number between f(a) and f(b)

Corollary of Intermediate Value Theorem is a theorem used to verify whether a function has at lease a root or not.

Assume:

• f(x) is continuous on the [a,b]
• f(a) and f(b) has opposite signs

then the equation f(x) = 0 has at least one root in the open interval (a, b).

This is a theorem that can be used for one variable.

If f(x) is continuous on [a, b], then f(x) takes on a least value m and a greatest value M on the interval.

Discontinuous function
without maximum value

If f(x) is continuous at point c and

     f(c) > 0

then there is a positive number t such that, whenever

     c - t < x < c + t

then

     f(x) > 0.