Continuity on an Interval

Definition

Continuity on an interval refers to the property of a function where the function's values change in a smooth, uninterrupted manner over a specified range of input values. This means that the function has no abrupt jumps or breaks within the given interval, and the function's graph can be drawn without lifting the pencil from the paper.

5 Must Know Facts For Your Next Test

1. A function is continuous on an interval if it is continuous at every point in that interval.
2. Continuity on an interval ensures that small changes in the input of the function result in small changes in the output of the function.
3. The Intermediate Value Theorem states that if a function is continuous on a closed interval $[a, b]$, and $f(a) \neq f(b)$, then the function must take on every value between $f(a)$ and $f(b)$.
4. Piecewise functions can be continuous on an interval if the function is continuous at the points where the pieces are joined.
5. The composition of continuous functions is also a continuous function.

Review Questions

• Explain the significance of a function being continuous on an interval.
  • A function being continuous on an interval means that the function's values change in a smooth, uninterrupted manner over the given range of input values. This ensures that small changes in the input result in small changes in the output, which is an important property for many applications, such as modeling real-world phenomena. Continuity on an interval also allows for the application of important theorems, such as the Intermediate Value Theorem, which states that a continuous function must take on every value between its minimum and maximum values on a closed interval.
• Describe how the composition of continuous functions affects the continuity of the resulting function.
  • The composition of continuous functions is also a continuous function. This means that if $f(x)$ and $g(x)$ are both continuous functions, then the composite function $h(x) = f(g(x))$ is also continuous. This property is important in many areas of mathematics, such as in the study of limits, derivatives, and integrals, where function composition is a common operation. The continuity of the composite function ensures that small changes in the input of the outer function $f(x)$ result in small changes in the output of the overall composition $h(x)$.
• Analyze the relationship between continuity and the Intermediate Value Theorem, and explain how this theorem can be used to determine the existence of solutions to equations.
  • The Intermediate Value Theorem states that if a function $f(x)$ is continuous on a closed interval $[a, b]$, and $f(a) \neq f(b)$, then the function must take on every value between $f(a)$ and $f(b)$. This theorem is closely tied to the concept of continuity on an interval, as it relies on the function's values changing in a smooth, uninterrupted manner. The Intermediate Value Theorem can be used to determine the existence of solutions to equations of the form $f(x) = c$, where $c$ is a constant, by finding an interval $[a, b]$ such that $f(a) < c < f(b)$ or $f(a) > c > f(b)$. Since the function is continuous on the interval, it must take on the value $c$ somewhere within the interval, guaranteeing the existence of a solution.

Related terms

Discontinuity:

A discontinuity is a point where a function is not continuous, meaning the function has a jump, a break, or is undefined at that point.

Limit:

The limit of a function at a point is the value that the function approaches as the input gets arbitrarily close to that point, but does not necessarily equal the function's value at that point.

Function Composition:

Function composition is the process of combining two or more functions to create a new function, where the output of one function becomes the input of the next function.