11.9 Continuity and Differentiability If a rock is dropped from a bridge which is 100 feet above water level, then the motion of the rock is continuous since the rock will pass through every altitude between 100 and 0 feet before it hits the water. In mathematics we use the phrase continuous function in a similar manner. A function whose graph has no breaks, holes, or vertical asymptotes is called a continuous function, since its graph proceeds uninterrupted. A function f is continuous at a number c if the following conditions are satisfied: i.
f  c  is defined ii.
lim f ( x) exists x c
iii.
lim f ( x)  f (c) x c
To illustrate the definition of continuity, the graph of each function shown below is not continuous at the number c. For the first function lim f ( x) does not exits , for the second f (c) is not defined, and for the third x c
function lim f ( x)  f (c) . If one or more of the three conditions stated in the definition are not x c
satisfied, we say f is discontinuous at c, or that f has a discontinuity at c. There are three types of discontinuities; removable, jump, and infinite. The first function has a jump discontinuity at c, while the other two functions have a removable discontinuity at c, since we could remove these discontinuities by defining the function value f  c  appropriately. An infinite discontinuity occurs where the function has a vertical asymptote, such as the function shown below. 1. Find all points of discontinuity for the function shown below. The function is discontinuous at x  0,3, and 5 2. Graph the piecewise defined function and determine all points where the function is discontinuous.  x  3 if x  1

f ( x)   x 2  2 if  1  x  3 
11 if x  3

y  x3
x y
1 2
2 1
3 0
y  x2  2
x y
1 3
0 2
1 3
3 11
The function is discontinuous at x  1 Theorem i.
A polynomial function f is continuous at every real number c. ii.
A rational function f 
h
is continuous at every real number except those numbers c that g
cause division by zero. Examples: Find all values where the function is discontinuous 3. f ( x)  2 x 2  3 x  5 Solution: f is a polynomial function, so there are no values where the function is discontinuous. 4. f  x  
1
x  x  3
Solution: f is a rational function, so the values where f is discontinuous are x  0 and x  3 x2
x  x2
x2
x2
Solution: f  x   2

x  x  2  x  2  x  1
5. f  x  
2
f is a rational function, so the values where f is discontinuous are x  2 and x  1 since they cause division by zero. 6. f  x  
x7 Solution: The square root of a negative number is not a real number, so the function is discontinuous when x  7 . The derivative of a function does not exist wherever the function is discontinuous, has a corner point, or has a vertical tangent. X = 1 is a corner point x X = 2 is a corner point At x = 2 there is a vertical asymptote. Case 11 Price Elasticity of Demand In the business world we are always concerned with how a change in price will affect the demand for an item. One way to measure the sensitivity of changes in price to demand is by the ratio of percent change in demand to percent change in price. This ratio is called the price elasticity of demand and is denoted by E. Elasticity of Demand Let q  f ( p) , where q is the demand at a price p. The elasticity of demand is as follows: i.
Demand is inelastic if E  1 . (Inelastic means a percent change in price will result in a smaller percent change in demand) ii.
Demand is elastic if E  1 . (elastic means a percent change in price will result in a greater percent change in demand) iii.
Demand is unit elastic if E  1 . (unit elastic means a percent change in price is the same as a percent change in demand) p dq
Where E   
q dp
7. Acme Stationary sells pens for $30 each. The demand equation for annual sells of these pens is q  1000 p  70, 000 , where p is the price per pen a) Find the elasticity of demand at the current price. b) Find the elasticity of demand if the price is raised by 1/3. Is this a good idea? Solution a) E  

p dq

q dp
30
 1000   1000  30   70, 000 
E  .75 Inelastic
Since the elasticity of demand is inelastic a percent change in price will result in a smaller percent change in demand. For example a 10% increase in price will result in a 7.5% decrease in demand. b) A 1/3 increase in price will now cost the consumer $40 per pen. E

p dq

q dp
40
 1000   1000  40   70, 000 
E  1.33 Elastic
Since the elasticity of demand is elastic a percent change in price will result in a greater percent change in demand. For example a 10% increase in price will result in a 13.3% decrease in demand.