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Determine the values of x, if any, at which the function is discontinuous. At each number where f is discontinuous, state the condition(s) for continuity that are violated. (Select all that apply.)f(x) = |x − 1|The function f is discontinuous at x = 1 because f is not defined at x = 1.The function f is discontinuous at x = 1 because lim x → 1 f(x) does not exist.The function f is discontinuous at x = 1 because lim x → 1 f(x) exists, but this limit is not equal to f(1).The function f is continuous everywhere because the three conditions for continuity are satisfied for all values of x.

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Answer:

The function f is continuous everywhere because the three conditions for continuity are satisfied for all values of x.

Step-by-step explanation:

The definition of continuity of a function is:

A function f (x) is continuous at x = a if the following three conditions are met:

  1. f(a) exists
  2. [tex]\lim_{x \to a} f(x)[/tex] exists
  3. [tex]\lim_{x \to a} f(x)[/tex] = f(a)

We check each condition when x=1

1) f(x)=|x-1| so f(1)=|1-1|=0

The value of f(x) exits when x=1

2) The limit as x approaches 1 of |x-1| can be checked with lateral limits:

[tex]\lim_{x \to 1^{-}} |x-1|[/tex]=0

[tex]\lim_{x \to 1^{+}}|x-1|[/tex]=0

Both limits have the same value to the limit exist when x approaches 1.

3) [tex]\lim_{x \to 1} |x-1|[/tex]=0=f(1)

The limit as x approaches 1 of f(x) is equal to the value of f(x) exits when x=1

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The function f is continuous everywhere because the three conditions for continuity are satisfied for all values of x.

Given that,

Function f(x) = | x-1 |

The function f is discontinuous at x = 1 because f is not defined at x = 1.

The function f is discontinuous at x = 1 because limx→1 f(x) does not exist.

The function f is discontinuous at x = 1 because limx→1 f(x) exists, but this limit is not equal to f(1).

We have to find,

The function f is continuous everywhere because the three conditions for continuity are satisfied for all values of x.

According to the question,

The definition of continuity of a function is:

A function f (x) is continuous at x = a if the following three conditions are met:

  • f(a) exists
  • [tex]lim_x_-_a = f(x)[/tex] exists
  •   [tex]lim_x_-_a = f(a)[/tex]

We check each condition when x = 1

  •  f(x) = x-1| so f(1) = |1-1| = 0

         The value of f(x) exits when x=1

  • The limit as x approaches 1 of |x-1| can be checked with lateral limits:

       [tex]= lim_x_-_1- = |x-1| = 0\\= lim_x_-_1+ = |x-1| = 0[/tex]

Both limits have the same value to the limit exist when x approaches 1.

  • The function f is discontinuous at x = 1 because limx → 1 f(x) exists, but this limit is not equal to f(1).

        [tex]= lim_x_-_1 = |x-1|[/tex] = 0 = f(1)

The limit as x approaches 1 of f(x) is equal to the value of f(x) exits when x=1.

Hence, The function f is continuous everywhere because the three conditions for continuity are satisfied for all values of x.

For more information about Limits click the link given below.

https://brainly.com/question/18008819

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