Enc1102

Section 1.5: Continuity
A graphical approach to continuity,
a) f(x) is continuous every where but x=c. We call this a removable (hole) discontinuity.


b) f(x)is not continuous at x=c. We call this a non-removable (asymptotic) discontinuity.

c) “JUMPS” will only occur when we have a piecewise function. (Function withdifferent formulas depending on the value of “x”.

Ex:

“JUMPS” will only occur when we have a piecewise function. (Function with different formulas depending on the value of “x”.

Ex:

f(x)is not continuous at x=c. We call this a Jump discontinuity.

d) We say f(x) is continuous at x=c


Definition of continuity at a point:
A function is said to be continuousat provided the following conditions are satisfied:
1. Exist (is defined)
2. Exist
3.

Example of discontinuities:
a.

b.

Definition 2: Continuous on an IntervalA function is continuous on an interval if it is continuous at EVERY point of the interval….there are no discontinuities at any point in the interval.

Definition 3: A function is said to becontinuous on a closed interval if the following conditions are satisfied:
1. is continuous on.
2. is continuous from the right of .

3. is continuous from the left of .

Properties ofContinuous Functions
If the functions f and g are continuous at x = c, then the following combinations are also continuous at x = c.

1. Sums: f + g
2. Difference f – g
3.Product: fg
4. Quotients: f/g provided g(c) is not 0
5. Compositions g(f(x)) provided f(x) is in the domain of g
6. Powers: fr/s, provided it is defined onan open interval containing c, where r and s are integers.
FACTS:
* Polynomials are continuous functions.

* Rational function is continuous at every point where the denominator is nonzero...