How Do You Prove That A Function Is Continuous At Some Point?

Health insurance, taxes and many consumer applications result in a models that are piecewise functions.  To prove these functions are continuous at some point, such as the locations where the pieces meet, we need to apply the definition of continuity at a point.

A function f is continuous at a point x = a if each of the three conditions below are met:

i.  f (a) is defined

ii.  \displaystyle \underset{x\to a}{\mathop{\lim }},f(x) is defined

iii.  \displaystyle \underset{x\to a}{\mathop{\lim }},f(x)=f(a)

In the problem below, we ‘ll develop a piecewise function and then prove it is continuous at two points.

Problem A company transports a freight container according to the schedule below.

  • First 200 miles is $4.00 per mile
  • Next 300 miles is $3.00 per mile
  • All miles over 500 is $2.50 per mile

Let C(x) denote the cost to move a freight container x miles.

a. Find a piecewise function for C(x).

For this function, there are three pieces. The first piece corresponds to the first 200 miles. The second piece corresponds to 200 to 500 miles, The third piece corresponds to miles over 500.

The board below show the function.


Let’s break this down a bit. In the first section, each mile costs $4.50 so x miles would cost 4.5x.

In the second piece, the first 200 miles costs 4.5(200) = 900. All miles over 200 cost 3(x-200). This gives the sum in the second piece.

In the third piece, we need $900 for the first 200 miles and 3(300) = 900 for the next 300 miles. In addition, miles over 500 cost 2.5(x-500).

b. Prove that C(x) is continuous over its domain.

Each piece is linear so we know that the individual pieces are continuous. However, are the pieces continuous at x = 200 and x = 500?

Let’s look at each one sided limit at x = 200 and the value of the function at x = 200.

m212_cont_2_bSince these are all equal, the two pieces must connect and the function is continuous at x = 200. At x = 500,

m212_cont_2_cso the function is also continuous at x = 500.

This means that the function is continuous for x > 0 since each piece is continuous and the  function is continuous at the edges of each piece.