Problem 28 - Differential Equations and Linear Algebra

Question

Sketch the given function and determine whether it is piecewise continuous on \([0, \infty)\). $$f(t)=n, \quad n \leq t < n+1, \quad n=0,1,2, \ldots$$

Step-by-Step Solution

Answer

The given function can be visualized as a staircase-like graph with constant values for each interval \([n, n+1)\), where \(n\) is an integer value. It is continuous on each individual interval but has jump discontinuities at the integer values. Therefore, the function is piecewise continuous on the interval \([0, \infty)\).

Step 1: Break down the function for different intervals

Let's first break down the given function into several intervals, based on the parameter \(n\): - For \(n=0\), the function is defined as \(f(t)=0,\) for \(0\leq t<1\). - For \(n=1\), the function is defined as \(f(t)=1,\) for \(1\leq t<2\). - For \(n=2\), the function is defined as \(f(t)=2,\) for \(2\leq t<3\). - This pattern continues for all integer values of \(n\).

Step 2: Sketch the graph of the function

Now that we have defined the function for each interval, let's sketch the graph on the given domain \([0, \infty)\): - On the interval \([0, 1)\), the function is constant at \(f(t)=0\). - On the interval \([1, 2)\), the function is constant at \(f(t)=1\). - On the interval \([2, 3)\), the function is constant at \(f(t)=2\). - And so on... We end up with a staircase-like graph, where each step has a width of \(1\) and a height equal to \(n\).

Step 3: Determine the continuity of the function

To determine if the function is piecewise continuous, we need to check its continuity for each interval: - On \([0, 1)\), the function is continuous since it's constant. - On \([1, 2)\), the function is continuous since it's constant. - On \([2, 3)\), the function is continuous since it's constant. So, the function is continuous on each individual interval. However, we also need to check continuity at the endpoints of each interval, these are the integer values (\(n=0, 1, 2, ...\)). At these points, there is always a jump (discontinuity) equal to \(1\), as the function jumps from the value \(n-1\) to \(n\) when moving from left to right along the domain.

Step 4: Conclusion

The given function is a staircase-like graph with constant values for each interval \([n, n+1)\), where \(n\) is an integer value. The function is continuous on each individual interval, but there are jump discontinuities at the integer values. Therefore, the function is piecewise continuous on the interval \([0, \infty)\).

Key Concepts

Staircase Function

A staircase function is a type of piecewise function that resembles a set of steps. Each "step" in the graph is a horizontal line segment that maintains a constant value over a specific interval, much like the steps of a staircase. The given function, which is defined as \( f(t) = n \) for \( n \leq t < n+1 \), demonstrates this perfectly:

• For each integer \( n \), the function maintains a constant output \( n \) within the interval \([n, n+1)\).
• The width of each step is uniform and spans exactly 1 unit.
• The height of each step increases incrementally by 1 as \( n \) increases.

The repeated pattern of steps creates a graph that bears a resemblance to a staircase, hence its name. These functions are useful for modeling situations where a quantity changes at discrete points, but remains constant between these points, like postal charges or tax brackets.

Interval Continuity

Interval continuity refers to the property of a function being uninterrupted and smooth over a particular range. In the case of our staircase function, the intervals \([n, n+1)\) for each integer \( n \) are where the function maintains continuity. Within these intervals,

• The function value does not change, remaining constant at \( n \).
• No sudden jumps or breaks occur, as the function value remains fixed.

This continuity within each segment is essential for understanding piecewise continuous functions as it highlights that different pieces can be independently continuous. However, the overall continuity of a function is determined by not only its behavior within intervals but also at the junctions or endpoints of these intervals.

Discontinuity at Endpoints

While the staircase function is continuous within each interval, it exhibits discontinuity at the endpoints, particularly at each integer \( n \). These are the points where the horizontal line segments "jump" to a new level:

• As we move from \( t = n-1 \) to \( t = n \), the function value shifts from \( n-1 \) to \( n \).
• Each jump represents a discontinuity, as there is no smooth transition between the two values.
• Discontinuities at these integer points define the piecewise nature of the entire structure.

This type of jump is typical in piecewise functions that model processes involving sudden changes. Understanding these discontinuities is crucial since they can impact analyses or calculations involving the integral or derivative of the function, demarcating where smooth equations break down.