Problem 44 - Calculus

Question

give an example of: A function that is increasing but not continuous on [0,10]

Step-by-Step Solution

Answer

An example is a piecewise function: \( f(x) = x \) for \( x \neq 5 \) and \( f(5) = 6 \).

Step 1: Understand the Concepts

To solve this exercise, we need to understand two concepts: what it means for a function to be increasing and what it means for it to be not continuous. An increasing function is one in which, for any two numbers \( x_1 \) and \( x_2 \) within the domain, if \( x_1 < x_2 \), then \( f(x_1) \leq f(x_2) \). A function is not continuous if there is at least one point in its domain where a limit does not equal the function's value.

Step 2: Choose an Appropriate Example

We'll use a piecewise function as our example. Consider the function \( f(x) \) defined on the interval \([0, 10]\) by: \[ f(x) = \begin{cases} x & \text{if } 0 \leq x < 5 \ x + 1 & \text{if } x = 5 \ x & \text{if } 5 < x \leq 10 \end{cases} \]This function increases at every point on the interval but is not continuous at \( x = 5 \), where there is a jump.

Step 3: Verify the Function is Increasing

To verify that \( f(x) \) is increasing, examine each case:- On \( [0, 5) \), since \( f(x) = x \), it clearly increases because as \( x \) increases, \( f(x) \) increases.- At \( x = 5 \), \( f(x) = 6 \) and on \( (5, 10] \), \( f(x) = x \). Here, \( 6 = f(5) \) which is greater than \( f(x) = x \) for any value \( x < 5 \).- On \( (5, 10] \), \( f(x) = x \) is clearly increasing as \( x \) increases.

Step 4: Verify the Function is Not Continuous

To show that \( f(x) \) is not continuous at \( x = 5 \), note that as \( x \to 5^- \), \( f(x) = 5 \), but \( f(5) = 6 \). There is a jump discontinuity at \( x = 5 \) because the limit from the left does not equal \( f(5) \).

Key Concepts

Increasing Functions

An increasing function is one that continuously rises as the input values go up. To put it simply, if you select any two points, say \( x_1 \) and \( x_2 \) within the function's domain, and \( x_1 \) is less than \( x_2 \), then it should follow that the output for \( x_1 \), denoted \( f(x_1) \), is less than or equal to the output for \( x_2 \), denoted \( f(x_2) \). This definition is reflected across any interval within the function's domain.

It's important to note that the term "increasing" does not necessarily imply a smooth progression without any plateaus. A function can be increasing even if the outputs for some intervals remain constant. But as soon as it dips, it causes the function to not be considered increasing anymore. A common practical example might be a function representing cumulative income over time, which rises as more income is earned, reflecting that the function is increasing.

Jump Discontinuity

Jump discontinuity occurs when there's an abrupt change in the function's value at a specific point, which causes a "jump" in the graph. Mathematically, this is seen when the left-hand limit and the right-hand limit differ at a certain point. The function suddenly increases or decreases without covering a smooth path.

Imagine you're walking on a path that suddenly has a step; you have to "jump" from one level to the next. Similarly, in the function defined in the exercise, there is such a jump at \( x = 5 \). As you approach this point from the left, the function heads towards the value \(5\), but instead, at \( x = 5 \), it suddenly jumps to \(6\). Hence, despite the function being increasing overall, the jump creates a break in continuity, making it discontinuous at \( x = 5 \).

Understanding jump discontinuities is essential for working with piecewise functions since these functions often define rules that cause such abrupt changes in values at certain critical points.

Piecewise Functions

Piecewise functions are those defined by multiple sub-functions, each applying to a certain interval within the main function's domain. These can be quite handy when a function behaves differently in different ranges. Imagine a situation where you calculate a taxi fare that charges a flat rate up to a certain distance and then varies based on additional kilometers.

In the exercise provided, the piecewise function is expressed over the interval \([0, 10]\) as follows:

• For \(0 \leq x < 5\), the function behaves as \(f(x) = x\).
• At \(x = 5\), it jumps to \(f(x) = x + 1 = 6\).
• For \(5 < x \leq 10\), it returns to \(f(x) = x\).

Notice how conditions define which parts of the function apply depending on \(x\)'s value. This structured approach offers a perfect solution to problems requiring different operations over one domain, leading to potential discontinuities as seen with jumps.