Question

How do you use the (analysis) definition of continuity to prove the following function `f(x) = 3x +5` is continuous for all x in R?

Answer 1

We will use the Epsilon-Delta definition of continuity for `f(x) = 3x + 5.` See answer below.

Explanation:

Epsilon-Delta definition of continuity on all `x_0 in RR` :

`AA x_0 in RR, AA epsilon > 0 EE delta_(epsilon, x_0) " such that" AA x in RR :`

`|x - x_0| < delta_(epsilon, x_0) rArr |f(x) - f(x_0)| < epsilon.`

So we must find `delta_(epsilon, x_0)` with respect to `epsilon` and `x_0` so that the implication will be true.

`|f(x) - f(x_0)| = |(3x + 5) - (3x_0 + 5)| = |3x - 3x_0| = |3(x - x_0)| = 3|x - x_0| < 3*delta_(epsilon, x_0) = epsilon` if we set down `delta_(epsilon, x_0) = epsilon/3.`

Therefore, `f(x)` is continuous `AA x_0 in RR.`

Notice that our `delta` only depends on `epsilon` and not on `x_0`, we call that uniform continuity.