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So how can we decide if a function is continuous at a particular number? We can check three different conditions. Let’s use the function y = f ( x ) represented in [link] as an example.

Graph of an increasing function with a discontinuity at (a, f(a)).

Condition 1 According to Condition 1, the function f ( a ) defined at x = a must exist. In other words, there is a y -coordinate at x = a as in [link] .

Graph of an increasing function with a discontinuity at (a, 2). The point (a, f(a)) is directly below the hole.

Condition 2 According to Condition 2, at x = a the limit, written lim x a f ( x ) , must exist. This means that at x = a the left-hand limit must equal the right-hand limit. Notice as the graph of f in [link] approaches x = a from the left and right, the same y -coordinate is approached. Therefore, Condition 2 is satisfied. However, there could still be a hole in the graph at x = a .

Condition 3 According to Condition 3, the corresponding y coordinate at x = a fills in the hole in the graph of f . This is written lim x a f ( x ) = f ( a ) .

Satisfying all three conditions means that the function is continuous. All three conditions are satisfied for the function represented in [link] so the function is continuous as x = a .

Graph of an increasing function with filled-in discontinuity at (a, f(a)).
All three conditions are satisfied. The function is continuous at x = a .

[link] through [link] provide several examples of graphs of functions that are not continuous at x = a and the condition or conditions that fail.

Graph of an increasing function with a discontinuity at (a, f(a)).
Condition 2 is satisfied. Conditions 1 and 3 both fail.
Graph of an increasing function with a discontinuity at (a, 2). The point (a, f(a)) is directly below the hole.
Conditions 1 and 2 are both satisfied. Condition 3 fails.
Graph of a piecewise function with an increasing segment from negative infinity to (a, f(a)), which is closed, and another increasing segment from (a, f(a)-1), which is open, to positive infinity.
Condition 1 is satisfied. Conditions 2 and 3 fail.
Graph of a piecewise function with an increasing segment from negative infinity to (a, f(a)) and another increasing segment from (a, f(a) - 1) to positive infinity. This graph does not include the point (a, f(a)).
Conditions 1, 2, and 3 all fail.

Definition of continuity

A function f ( x ) is continuous at x = a provided all three of the following conditions hold true:

  • Condition 1: f ( a ) exists.
  • Condition 2: lim x a f ( x ) exists at x = a .
  • Condition 3: lim x a f ( x ) = f ( a ) .

If a function f ( x ) is not continuous at x = a , the function is discontinuous at x = a .

Identifying a jump discontinuity

Discontinuity can occur in different ways. We saw in the previous section that a function could have a left-hand limit    and a right-hand limit    even if they are not equal. If the left- and right-hand limits exist but are different, the graph “jumps” at x = a . The function is said to have a jump discontinuity.

As an example, look at the graph of the function y = f ( x ) in [link] . Notice as x approaches a how the output approaches different values from the left and from the right.

Graph of a piecewise function with an increasing segment from negative infinity to (a, f(a)), which is closed, and another increasing segment from (a, f(a)-1), which is open, to positive infinity.
Graph of a function with a jump discontinuity.

Jump discontinuity

A function f ( x ) has a jump discontinuity    at x = a if the left- and right-hand limits both exist but are not equal: lim x a f ( x ) lim x a + f ( x ) .

Identifying removable discontinuity

Some functions have a discontinuity, but it is possible to redefine the function at that point to make it continuous. This type of function is said to have a removable discontinuity. Let’s look at the function y = f ( x ) represented by the graph in [link] . The function has a limit. However, there is a hole at x = a . The hole can be filled by extending the domain to include the input x = a and defining the corresponding output of the function at that value as the limit of the function at x = a .

Graph of an increasing function with a removable discontinuity at (a, f(a)).
Graph of function f with a removable discontinuity at x = a .

Removable discontinuity

A function f ( x ) has a removable discontinuity    at x = a if the limit, lim x a f ( x ) , exists, but either

  1. f ( a ) does not exist or
  2. f ( a ) , the value of the function at x = a does not equal the limit, f ( a ) lim x a f ( x ) .

Practice Key Terms 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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