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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 restricte the values of x until we got what we wanted, making sure that 
    • x got the same distance away from c to 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 boys rejoice.

In symbols,

  • We are given a continuous function f and a value c.
  • We pick a real number ε>0 (epsilon) that said how far we wanted to let f(x) move away from f(c).
  • We restrict the values of x until we got what we wanted, 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 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, sorry, that's the recipe for Nestle Tollhouse Cookies.

  • Write down the inequality f(c) - ε < f(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 -ε < f(x) - f(c) < ε which is the same thing as saying f(c) - ε < f(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, we find what we want. In symbols, we say |c| < δ which means the same thing as -δ, which means the same thing as - δ.



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