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
Mar 5, 2016

Answer:

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.