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.
- 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.
- 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 |x - 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.
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 |x - c| < δ.