Continuity

The  exercise appears under the Differential calculus Math Mission. This exercise explores the idea of continuity by the limit definition.

Types of Problems
There are three types of problems in this exercise:


 * 1) Find the limit from the graph: This problem takes a graph with several discontinuities. The student is asked to select which x-value satisfies the conditions of continuity described at the beginning of the problem.Continuity1.png
 * 2) Find the value to make the function continuous: This problem has a function with a removable discontinuity. The student is asked how the function should be defined to make the function continuous at this removable discontinuity.Continuity2.png
 * 3) Use algebra to make the piecewise function continuous: This problem describes a function with discontinuities and unknown coefficients in places. The student is asked to use algebra to find the values that can make the function continuous.Continuity3.png

Strategies
Knowledge of limits, systems of equations and the concept of continuity are encouraged to ensure success on this exercise.
 * 1) Continuity informally means that one can draw the graph without lifting the paper off the pencil.
 * 2) Dicontinuities are removable when they leave a dot in the graph. One fills in the dot.
 * 3) Discontinuities are unremovable if they are at asymptote or a jump.
 * 4) On Use algebra to make the piecewise function continuous some problem require creating and solving a system of equations with multiple unknowns.

Real-life Applications

 * 1) Limits are used to define both the derivative and the integral.
 * 2) The concept of infinitesimals (arbitrarily close to) has applications to anything where precise answers are not always practical or possible.