Limits and continuity

Continuity and discontinuities

Explore what happens near x = 0. Compare the value at the point with the values approached from the left and right.

At an interior point a, continuity requires three things: f(a) is defined, both sides approach the same finite value, and that value equals f(a). This approaching value is called the limit; we will study it next. At a domain endpoint, use the side inside the domain.

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1. A connected graph

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Example graph
f(x)=x+1
-4-4-2-22244xy

For f(x) = x + 1, both sides approach 1 and f(0) = 1. The function is continuous at zero. Move the distance slider toward zero and compare the two outputs.

From the left
f(-1) = 0
At x = 0
f(0) = 1
From the right
f(1) = 2

Press Enter to check your answer.

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A filled point belongs to the graph. An open circle marks an excluded point. Dashed guides mark the two sampled inputs. The graph shows only −5 ≤ y ≤ 5. Values outside this window still appear in the table.

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2. A removable hole

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Example graph
f(x)=\frac{x(x+1)}{x},\quad x\ne0
-4-4-2-22244xy

For x ≠ 0, x(x + 1)/x equals x + 1. But cancelling x does not restore the excluded input zero. Both sides approach 1 while f(0) is undefined. Defining f(0) = 1 fills the hole and makes the extended function continuous.

From the left
f(-1) = 0
At x = 0
f(0) = Undefined
From the right
f(1) = 2

Press Enter to check your answer.

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A filled point belongs to the graph. An open circle marks an excluded point. Dashed guides mark the two sampled inputs. The graph shows only −5 ≤ y ≤ 5. Values outside this window still appear in the table.

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3. A jump

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Example graph
f(x)=\begin{cases}1 & x<0\\2 & x\ge0\end{cases}
-4-4-2-22244xy

Here f(x) = 1 for x < 0 and f(x) = 2 for x ≥ 0. The left side approaches 1; the right side approaches 2. Defining a different value at zero cannot join these two sides.

From the left
f(-1) = 1
At x = 0
f(0) = 2
From the right
f(1) = 2

Press Enter to check your answer.

A filled point belongs to the graph. An open circle marks an excluded point. Dashed guides mark the two sampled inputs. The graph shows only −5 ≤ y ≤ 5. Values outside this window still appear in the table.

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4. Unbounded behavior

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Example graph
f(x)=\frac1x,\quad x\ne0
-4-4-2-22244xy

For f(x) = 1/x, approaching zero from the right gives larger positive values; approaching from the left gives negative values with increasing magnitude. Neither side approaches a finite value. No single value at zero repairs this behavior. The vertical line x = 0 is an asymptote.

From the left
f(-1) = -1
At x = 0
f(0) = Undefined
From the right
f(1) = 1

Press Enter to check your answer.

Open in worksheet

A filled point belongs to the graph. An open circle marks an excluded point. Dashed guides mark the two sampled inputs. The graph shows only −5 ≤ y ≤ 5. Values outside this window still appear in the table.

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