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At a Glance - The Formal Version

When we graph continuous functions, three things happen:

  • We are given a continuous function f and a value c.
  • We decide how far we wanted to let f(x) move away from f(c).
  • We restrict the values of x until we get what we want, making sure that 
    • x is the same distance away from c on either side, and that
    • we didn't restrict x to just equal c, since then we would find a dot.

Enter the Greek letters—frat bros rejoice.

In symbols,

  • We're given a continuous function f and a value c.
  • We pick a real number ε > 0 (epsilon) that specifies how far we want to let f(x) move away from f(c).
  • We restrict the values of x until we get what we want, ending with c δ

The continuous function guarantee says that no matter what ε > 0 we pick, we'll be able to find a δ > 0.

We use the Greek letters to specify how close various things are to one another. We use the letter ε to specify how close we want f(x) and f(c) to be. The definition of continuity says for any ε we can find an appropriate δ such that if x is within δ of c, we can find our desired value.

We have a great recipe for cooking up δ with the ingredients f, c, and ε. First we combine flour, baking soda, and salt in a bowl, then we...no wait. That's the recipe for Nestle Tollhouse Cookies.

  • Write down the inequality f(c) - εf(x) < f(c) + ε and fill in whatever we are given for c, f, and ε.
      
  • Solve the inequality for x.
      
  • Subtract c from all parts of the inequality and find δ.

The super-formal definition of continuity says:

The function f is continuous at c if for any real ε > 0 there exists a real δ > 0 such that if |c| < δ, then |f(x) - f(c)| < ε.

To translate, if f is continuous at c we can pick any real ε > 0 and say we want to have f(x) and f(c) within ε of each other. In symbols, we write this |f(x) - f(c)| < ε.

This is the same thing as saying -εf(x) – f(c) < ε which is the same thing as saying f(c) – εf(x) <  f(c) + ε.

In pictures,

Since f is continuous, we have a guarantee that we can find some real δ > 0 such that if x is within δ of c, f(x) is within ε of f(c). In symbols, we say |c| < δ.

But as always, a picture is worth a thousand Greek symbols.

Example 1

Let f(x) = 2x, c = 4, and ε = 0.5. Find an appropriate δ > 0, so that if x is within δ of 4, f(x) will be within ε of f(c).


Example 2

Let f(x) = x2, c = 2, and ε = 0.1. What is an appropriate value of δ > 0, that will guarantee that if |xc| < δ, |f(x) – f(c)| < ε?


Example 3

Let f(x) = 4x + 1. 

Find a value of δ such that if |x – 5| < δ, then |f(x) – f(5)| < 0.1.


Example 4

Let f(x) = sin(x

Find a value of δ such that if , then .


Example 5

Let . Find a value of δ such that if |x – 2| < δ, then |f(x) – 0.5| < 0.005.


Example 6

Let . What is a value of δ that will guarantee that if |x – 7| < δ, then ?


Exercise 1

For the given continuous function, value of c, and ε, what is f(c) and an appropriate δ that will guarantee the continuity of f?

5x – 2 , c = 3 , and ε = 1.


Exercise 2

For the given continuous function, value of c, and ε, find f(c) and a value of δ such that if |xc| < δ, |f(x) – f(c)| < ε.

ex , c = 0 , and ε = 1.


Exercise 3

For the given continuous function, value of c, and ε, what is f(c) and a value of δ guaranteed by the continuity of f?

x2 + 2 , c = -2 , and ε = 0.2.


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