Continuity and Discontinuity

Over the weekend, our homework was to complete a survey evaluating limits through the topic of roads and bridges. On Monday, we began our discussion on continuity by thinking in these roads and bridges terms:

Continuity- a function f(x) is continuous at x=a if

1. the roads meet
2. there is a bridge
3. the roads meet at the bridge

After conceptually understanding what continuity is, we decided to translate this definition into calculus lingo.

Continuity- a function f(x) is continuous at x=a if

1. the limit of x as it approaches a exists
2. f(a) exists
3. the limit of x as it approaches a is equal to f(a)

In other words

• If you can draw it without lifting your pencil, it’s continuous.
• f(x) is continuous at x=a if
• the limit from the left equals the limit from the right (or the roads come to the same spot)
• a y-value exists at x=a (bridge exists; there isn’t a hole or asymptote there)
• the y-value, the limit from the left, and the limit from the right are all the same number

Discontinuity

• A function has a point of discontinuity at x=a if it fails one or more of the four criteria listed below

The Four Types of Discontinuities and How to Find Them:

1. Removable
1. Piecewise function
2. Hole (a factor cancels)
2. Jumps
1. Absolute value division function
2. Greatest integer function
3. Piecewise function
3. Infinite
1. y=tan(x) (infinite discontinuities at odd integers of pi/2)
2. Reciprocal (vertical asymptote- cannot cancel an x in the denominator)
4. Restricted
1. Square root (continuous for all x-values in its domain, but not outside its domain)
2. Logarithms (only continuous for x>0, unless it is transformed)

The worksheet for discontinuity:

Based on the definition of continuity and the four types of discontinuity, we were able to solve the following problems in class:

To find a value that makes a function continuous, plug in the the boundary (in this case, the boundary is 1) into the x of each part of the piecewise function and solve. Then, set up the answers from each part of the piecewise function and save for a, as shown above. 

Here are answers to the homework problems!

Differentiability

Differentiability: f(x) is not differentiable at x=a (f’(x) does not exist) when f’(x) is not continuous at x=a.

For example:

The Four Ways a Function can Fail to be Differentiable at x=a:

1. A corner
1. Different one sided derivatives causes the graph to jump (ex. absolute value graph)
2. A vertical tangent
1. One-sided derivatives go toward infinity (ex. y=x^⅓)
3. A cusp
1. One-sided derivatives go toward opposite infinities (one toward positive infinity and the other toward negative infinity)
4. Any type of discontinuity
1. Any of the four types of discontinuity explained in the previous section!

The worksheet for differentiability:

Remember:

• Continuity does not imply differentiability, but
• Differentiability does imply continuity.

In other words:

• f’(c)=lim x as it approaches c [f(x)-f(c)]/(x-c) must exist if f is differentiable at x=c

Based on these notes of differentiability, we were able to complete the following in class problems:

To make it differentiable, you find the derivative of the top part of the piecewise function and the derivative of the bottom of the piecewise function, set them equal to one another, and then solve. In this case, you also had to make it continuous, which is explained in a problem above. 

Here are the homework answers!

Reminder

Don’t forget to study for the assessment on tuesday!