Properties of continuous functions

  • 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 =c.
  • 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
      THEOREM 2.7.3
If the function f and g are continuous at c then
    - f + g is continuous at c;
    - f - g is continuous at c;
    - f.g is continuous at c;
    - f/g is continuous at c if g(c) ≠ 0 and is discontinuous at c if g(c)=0
      Proof
Let f and g be continuous function at the number c
limx → c f(x) = f(c)
limx → c g(x) = g(c)
limx → c [f(x).g(x)] = limx → c f(x).limx → c g(x) = f(c).g(c)
      Example
h(x) = (x2 - 9)/(x2 - 5x + 6) = f(x)/g(x)
h(x) will be continuous at all points c If g(c) ≠ 0
h(x) is continuous everywhere except at x = 2 and x = 3

(a) F is discontinuous at x = a

(b) F is discontinuous at x = b

(c) F is discontinuous at x = a, b
      DEFINITION
A function f(x) is called continuous from left at the point c if the conditions in the left column below are satisfied and is called continuous from the right at the point c if the conditions in the right column are satisfied.
f(c) is defined   f(c) is defined
limx → c- f(x) exists   limx → c+ f(x) exists
limx → c- f(x) = f(c)   limx → c+ f(x) = f(c)

      DEFINITION
A function f(x) is said to be continuous on a closed interval [a, b] if the following conditions are satisfied:
-f(x) is continuous on [a, b];
-f(x) is continuous from the right at a;
-f(x) is continuous from the left at b.

      Example
Show that f(x) is continuous on [-3, 3]
f(x) = √9 - x2
for c in (-3, 3)
limx → c f(x) = limx → c9 - x2 = √limx → c 9 - x2 = √9 - c2 = f(c)
So f(x) is continuous on (-3, 3)
Also
limx → 3- f(x) = limx → 3-9 - x2 = √limx → 3- 9 - x2 = 0 = f(3) limx → 3+ f(x) = limx → 3+9 - x2 = √limx → 3+ 9 - x2 = 0 = f(3)
So f(x) is continuous on [-3, 3]
      INTERMEDIATE-VALUE THEOREM
If f(x) is continuous on a closed interval [a, b] and c is any number between f(x) and f(b), inclusive, then there is at least one number x in the interval [a, b] such that f(x) = c

      THEOREM 2.7.10
If f(x) is continuous on [a, b], and if f(a) and f(b) have opposite signs, then there is at least one solution of the equation f(x) = 0 in the interval (a, b)

      Example
    x3 - x - 1 = 0
f(1) = -1         f(2) = 5
This equation cannot be solved readily by factoring because the left side has no simple factors.

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