One-Sided Continuity

Definition

One-sided continuity refers to the concept of a function being continuous from one side, either from the left or the right, at a particular point. This is an important consideration in the study of continuity, as a function may exhibit different behavior on either side of a point.

5 Must Know Facts For Your Next Test

1. A function is said to be one-sided continuous at a point if the function is continuous from either the left or the right at that point.
2. If a function is one-sided continuous at a point, it means that the left-hand limit or the right-hand limit of the function exists at that point.
3. One-sided continuity is an important concept in understanding the overall continuity of a function, as a function may be continuous from one side but not the other.
4. The distinction between left-hand and right-hand continuity is crucial in analyzing the behavior of a function near a particular point.
5. One-sided continuity is a weaker condition than two-sided continuity, where a function must be continuous from both the left and the right at a point.

Review Questions

• Explain the difference between left-hand continuity and right-hand continuity of a function.
  • Left-hand continuity refers to a function being continuous as the input approaches a point from the left, while right-hand continuity refers to a function being continuous as the input approaches the same point from the right. A function may be one-sided continuous, meaning it is continuous from either the left or the right, but not necessarily both. This distinction is important in understanding the overall continuity of a function at a particular point.
• Describe how one-sided continuity is related to the existence of limits.
  • For a function to be one-sided continuous at a point, it means that the limit of the function must exist from that particular side. If a function is left-hand continuous at a point, it implies that the left-hand limit of the function exists at that point. Similarly, if a function is right-hand continuous at a point, it means the right-hand limit of the function exists. The existence of these one-sided limits is a necessary condition for one-sided continuity.
• Analyze the relationship between one-sided continuity and two-sided continuity, and explain the significance of this distinction.
  • Two-sided continuity, where a function is continuous from both the left and the right at a point, is a stronger condition than one-sided continuity. If a function is two-sided continuous at a point, it means the function is continuous from both sides, and the left-hand and right-hand limits of the function are equal at that point. However, a function may be one-sided continuous, meaning it is continuous from either the left or the right, but not necessarily both. This distinction is important in understanding the overall behavior of a function and its continuity properties at specific points.

Related terms

Continuity:

Continuity is a property of a function where the function has no breaks or jumps in its graph, allowing it to be drawn without lifting the pencil.

Left-Hand Limit:

The left-hand limit of a function at a point is the value the function approaches as the input approaches the point from the left.

Right-Hand Limit:

The right-hand limit of a function at a point is the value the function approaches as the input approaches the point from the right.