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?

1 Answer
Shura
Mar 5, 2016

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.

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