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Determine whether the function f ( x ) = { 1 x , x 2 9 x 11.5 , x > 2 is continuous at x = 2.

yes

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Determining whether a rational function is continuous at a given number

Determine whether the function f ( x ) = x 2 25 x 5 is continuous at x = 5.

To determine if the function f is continuous at x = 5 , we will determine if the three conditions of continuity are satisfied at x = 5.

Condition 1:

f ( 5 )  does not exist . Condition 1 fails .

There is no need to proceed further. Condition 2 fails at x = 5. If any of the conditions of continuity are not satisfied at x = 5 , the function f is not continuous at x = 5.

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Determine whether the function f ( x ) = 9 x 2 x 2 3 x is continuous at x = 3. If not, state the type of discontinuity.

No, the function is not continuous at x = 3. There exists a removable discontinuity at x = 3.

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Determining the input values for which a function is discontinuous

Now that we can identify continuous functions, jump discontinuities, and removable discontinuities, we will look at more complex functions to find discontinuities. Here, we will analyze a piecewise function to determine if any real numbers exist where the function is not continuous. A piecewise function    may have discontinuities at the boundary points of the function as well as within the functions that make it up.

To determine the real numbers for which a piecewise function composed of polynomial functions is not continuous, recall that polynomial functions themselves are continuous on the set of real numbers. Any discontinuity would be at the boundary points. So we need to explore the three conditions of continuity at the boundary points of the piecewise function.

Given a piecewise function, determine whether it is continuous at the boundary points.

  1. For each boundary point a of the piecewise function, determine the left- and right-hand limits as x approaches a , as well as the function value at a .
  2. Check each condition for each value to determine if all three conditions are satisfied.
  3. Determine whether each value satisfies condition 1: f ( a ) exists.
  4. Determine whether each value satisfies condition 2: lim x a f ( x ) exists.
  5. Determine whether each value satisfies condition 3: lim x a f ( x ) = f ( a ) .
  6. If all three conditions are satisfied, the function is continuous at x = a . If any one of the conditions fails, the function is not continuous at x = a .

Determining the input values for which a piecewise function is discontinuous

Determine whether the function f is discontinuous for any real numbers.

f x = { x + 1 , x < 2 3 , 2 x < 4 x 2 11 , x 4

The piecewise function is defined by three functions, which are all polynomial functions, f ( x ) = x + 1 on x < 2 , f ( x ) = 3 on 2 x < 4 , and f ( x ) = x 2 5 on x 4. Polynomial functions are continuous everywhere. Any discontinuities would be at the boundary points, x = 2 and x = 4.

At x = 2 , let us check the three conditions of continuity.

Condition 1:

f ( 2 ) = 3 Condition 1 is satisfied .

Condition 2: Because a different function defines the output left and right of x = 2 , does lim x 2 f ( x ) = lim x 2 + f ( x ) ?

  • Left-hand limit: lim x 2 f ( x ) = lim x 2 ( x + 1 ) = 2 + 1 = 3
  • Right-hand limit: lim x 2 + f ( x ) = lim x 2 + 3 = 3

Because 3 = 3 , lim x 2 f ( x ) = lim x 2 + f ( x )

Condition 2 is satisfied .

Condition 3:

lim x 2 f ( x ) = 3 = f ( 2 ) Condition 3 is satisfied .

Because all three conditions are satisfied at x = 2 , the function f ( x ) is continuous at x = 2.

At x = 4 , let us check the three conditions of continuity.

Condition 2: Because a different function defines the output left and right of x = 4 , does lim x 4 f ( x ) = lim x 4 + f ( x ) ?

  • Left-hand limit: lim x 4 f ( x ) = lim x 4 3 = 3
  • Right-hand limit: lim x 4 + f ( x ) = lim x 4 + ( x 2 11 ) = 4 2 11 = 5

Because 3 5 , lim x 4 f ( x ) lim x 4 + f ( x ) , so lim x 4 f ( x ) does not exist.

Condition 2 fails .

Because one of the three conditions does not hold at x = 4 , the function f ( x ) is discontinuous at x = 4.

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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