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 lim_{x → 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
-lim_{x → c} f(x) exist
-lim_{x → 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
lim_{x → 2} g(x) = lim_{x → 2} (x^{2} - 4)/(x - 2) = lim_{x → 2} (x + 2) = 4
g(x) is discontinuous because
lim_{x → 2} g(x) ≠ g(2)
Example
f(x) = x^{2} - 2x + 1
lim_{x → c} f(x) = lim_{x → c} (x^{2} - 2x + 1)
f(x) = c^{2} - 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
lim_{x → c} p(x) = p(c)
Example
If c < 0
f(c) = -c
lim_{x → c} f(x) = lim_{x → 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
lim_{x → c} f(x) = f(c)
lim_{x → c} g(x) = g(c)
lim_{x → c} [f(x).g(x)] = lim_{x → c} f(x).lim_{x → c} g(x) = f(c).g(c)
Example
h(x) = (x^{2} - 9)/(x^{2} - 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 | |
lim_{x → c-} f(x) exists | lim_{x → c+} f(x) exists | |
lim_{x → c-} f(x) = f(c) | lim_{x → 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 - x^{2}
for c in (-3, 3)
lim_{x → c} f(x) = lim_{x → c} √9 - x^{2} = √lim_{x → c} 9 - x^{2} = √9 - c^{2} = f(c)
So f(x) is continuous on (-3, 3)
Also
lim_{x → 3-} f(x) = lim_{x → 3-} √9 - x^{2} = √lim_{x → 3-} 9 - x^{2} = 0 = f(3) lim_{x → 3+} f(x) = lim_{x → 3+} √9 - x^{2} = √lim_{x → 3+} 9 - x^{2} = 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
x^{3} - 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.