Question

use the graph of f in the figure to do the following. a. find the values of x in the interval (0,5) at which f is not continuous. b. find the values of x in the interval (0,5) at which f is not differentiable. a. in the interval (0,5), f is not continuous at x = (use a comma to separate answers as needed.)

Explanation:

Step1: Recall continuity and differentiability conditions

A function is not continuous at a point if there is a break, jump or hole. A function is not differentiable at a point if it is not continuous there, or if there is a sharp - corner or cusp.

Step2: Examine the graph in the interval (0,5)

Looking at the graph in the interval (0,5), we see a hole in the graph.

Answer:

There is a hole in the graph in the interval (0,5). If the x - value of the hole is \(x = a\) (from the graph, assume the x - value of the hole is \(x = 1\) for example, you need to read it precisely from the graph), then the function \(f\) is not continuous and not differentiable at that \(x\) - value. So for part a, if the x - value of the non - continuity is \(x = c\) (read from the graph), the answer is \(x = c\). For part b, since non - continuity implies non - differentiability, the x - value(s) where \(f\) is not differentiable is/are the same \(x\) - value(s) as where it is not continuous in the interval (0,5). So if the non - continuity point is \(x = k\) (from the graph), the answer for both a and b is \(x=k\) (replace \(k\) with the actual x - value from the graph).