- Mathematical definition of continuity of functions
- Properties of continuous functions
- Continuity of polynomials and rational functions
- Continuity of composite functions
- The intermediate value theorem
Continuity of a function becomes obvious from its graph
Discontinuous: as f(x) is not defined at x = c
Discontinuous: as f(x) has a gap at x = c.
Discontinuous: not defined at x = c.
Function has different functional and limiting values at x =
- f(x) is undefined at c
- The limx → c f(x) does not exist.
- Values of f(x) and the values of the limit differ at the point c
Definition
A function f(x) is said to be continuous at a point c if the following conditions are satisfied
-f(c) is defined
-limx → c f(x) exist
-limx → c f(x) = f(c)
- If f(x) is continuous at all points in an interval (a, b), then f(x) is continuous on (a, b)
- A function continuous on the interval (-∞; +∞) is called a continuous function
Example
- f(x) is discontinuous at x = 2 because f(2) is undefined
By definition of g g(2) = 3
limx → 2 g(x) = limx → 2 (x2 - 4)/(x - 2) = limx → 2 (x + 2) = 4
g(x) is discontinuous because
limx → 2 g(x) ≠ g(2)
Example
f(x) = x2 - 2x + 1
limx → c f(x) = limx → c (x2 - 2x + 1)
f(x) = c2 - 2c + 1
f(x) = f(c)
So, f is continuous at x = c
THEOREM 2.7.2
Polynomials are continuous functions
If P is polynomial and c is any real number then
limx → c p(x) = p(c)
Example
If c < 0
f(c) = -c
limx → c f(x) = limx → c |x| = -c
-x may be negative to begin with but since ot approaches c which is positive or 0, we use the first part of the definition of f(x) to evaluate the limit