Problem 43 - Calculus: Early Transcendentals

Question

Find the numbers at which \( f \) is discontinuous. At which of these numbers is \( f \) continuous from the right, from the left, or neither? Sketch the graph of \( f \). \( f(x) = \left\\{ \begin{array}{ll} x + 2 & \mbox{if \) x < 0 \(}\\\ e^x & \mbox{if \) 0 \le x \le 1 \(} \\ 2 - x & \mbox{if \) x > 1 \(} \end{array} \right.\)

Step-by-Step Solution

Answer

The function is discontinuous at \( x = 0 \) and \( x = 1 \). It is neither continuous from the left nor right at these points.

Step 1: Identify Points of Discontinuity

To find the points of discontinuity for the piece-wise function, examine where each of the pieces meet. This occurs at the boundaries of the intervals: at \( x = 0 \) and \( x = 1 \). Check whether the limits exist from both sides at these points and if they are equal to the value of the function at those points.

Step 2: Test Discontinuity at x = 0

Evaluate the limit as \( x \) approaches 0 from the left, \( \lim_{{x \to 0^-}} f(x) = \lim_{{x \to 0^-}} (x + 2) = 2 \).Compare this to the limit from the right, \( \lim_{{x \to 0^+}} f(x) = \lim_{{x \to 0^+}} e^x = 1 \).Since \( \lim_{{x \to 0^-}} f(x) eq \lim_{{x \to 0^+}} f(x) \), there is a discontinuity at \( x = 0 \).\Since the limit does not equal \( f(0) = e^0 = 1 \), the function is neither continuous from the left nor right at \( x = 0 \).

Step 3: Test Discontinuity at x = 1

Evaluate the limit as \( x \) approaches 1 from the left, \( \lim_{{x \to 1^-}} f(x) = \lim_{{x \to 1^-}} e^x = e \).Compare this to the limit from the right, \( \lim_{{x \to 1^+}} f(x) = \lim_{{x \to 1^+}} (2 - x) = 1 \).Since \( \lim_{{x \to 1^-}} f(x) eq \lim_{{x \to 1^+}} f(x) \), there is a discontinuity at \( x = 1 \).Since neither one-sided limit equals \( f(1) = e \), it is neither continuous from the left nor right at \( x = 1 \).

Step 4: Sketch the Graph

Draw a graph for the function:1. For \( x < 0 \), sketch the line \( y = x + 2 \).2. For \( 0 \le x \le 1 \), plot the exponential curve \( y = e^x \).3. For \( x > 1 \), sketch the line \( y = 2 - x \).Mark the discontinuities at \( x = 0 \) and \( x = 1 \) with open circles and plot the curves to show the behavior around these points.

Key Concepts

Piecewise Functions

A piecewise function is a type of mathematical function characterized by different rules or expressions for different parts of its domain. In simple terms, it behaves like a "stitch-together" of smaller functions, each valid on a specific interval. These functions are useful in real-world situations where a formula may not be consistent across the entire range, such as when modeling tax brackets or shipping costs.

The given function, for instance, is divided into three sections:

• For values of \( x < 0 \), the function operates as \( f(x) = x + 2 \).
• From \( x = 0 \) to \( x = 1 \), the exponential function \( f(x) = e^x \) takes over.
• For values greater than \( x = 1 \), the function used is \( f(x) = 2 - x \).

Understanding how these pieces fit together is essential in determining how the function behaves and whether it is continuous throughout its range.

Limits and Continuity

The concepts of limits and continuity are fundamental in calculus. They help describe how a function behaves near specific points, specifically at the boundaries between different pieces of a piecewise function.

To determine continuity, we check the limits of the function as \( x \) approaches a boundary (like \( x = 0 \) or \( x = 1 \)) from both the left and the right. If these one-sided limits agree and match the function's value at that point, the function is continuous there. Otherwise, it indicates a discontinuity.

For the given function,

• At \( x = 0 \), the left-hand limit is 2, and the right-hand limit is 1. Since these do not match, there is a discontinuity.
• At \( x = 1 \), the left-hand limit is \( e \), and the right-hand limit is 1, signaling another discontinuity.

These discontinuities mean the function is neither left nor right continuous at both points.

Graphical Representation of Discontinuity

Visualizing a function can often make it easier to understand where and how discontinuities occur. The graphical representation of a piecewise function shows the different parts of the function in their respective intervals and highlights where the function "jumps" or is discontinuous.

For the given function, create a graph with three distinct segments:

• The line \( y = x + 2 \) for \( x < 0 \)
• The curve \( y = e^x \) for \( 0 \le x \le 1 \)
• The line \( y = 2 - x \) for \( x > 1 \)

At \( x = 0 \) and \( x = 1 \), open circles indicate discontinuities. These points show where the value of the function does not align from one segment to the next. This graphical approach provides a clear illustration of why the function isn't continuous at these specific points.