Problem 5 - Calculus Early Transcendentals

Question

Sketch the graph of a function that is continuous everywhere except at \(x=3\) and is continuous from the left at \(3 .\)

Step-by-Step Solution

Answer

A piecewise function: \( f(x) = x \) for \( x < 3 \), \( f(x) = 3 \) at \( x = 3 \), and \( f(x) = x + 1 \) for \( x > 3 \).

Step 1: Identify Discontinuity

We need a function that is continuous everywhere except at a single point, which is specified as \(x=3\). This means there must be some form of discontinuity at \(x=3\).

Step 2: Choose a Function Type

A suitable choice for such functions is a piecewise function, which defines the function differently in different intervals while introducing a discontinuity at \(x=3\).

Step 3: Define the Left-side Continuity

Ensure the function is continuous from the left at \(x=3\). This means \(\lim\limits_{x \to 3^-} f(x) = f(3)\). Choose the left-side function to be a simple linear function, such as \(f(x) = x\), applicable for \(x < 3\). Set \(f(3) = 3\) to ensure left-side continuity.

Step 4: Define the Right-side of the Function

To introduce discontinuity at \(x=3\), the right-side function can be different from the left. Example: let \(f(x) = x + 1\) for \(x > 3\). Then, \(\lim\limits_{x \to 3^+} f(x) = 4\), creating a discontinuity as \(4 eq 3\).

Step 5: Sketch the Graph

On a graph, draw a straight line for \(f(x) = x\) on the left side of \(x=3\) including the point \((3,3)\). Draw a different line for \(f(x) = x+1\) starting just after \(x=3\), which does not include the point \((3,4)\). There should be an open circle at \((3,4)\) indicating the discontinuity point.

Key Concepts

Discontinuity

A discontinuity in a function occurs when there is a sudden break or jump in the graph of the function. In our problem, the discontinuity happens at the point where the curve of the function breaks at \(x=3\). This means the function is not smooth at this particular point. There are different types of discontinuities, such as:

• Jump Discontinuity: This occurs when the left-hand and right-hand limits at a point exist but are not equal. Our example has a jump discontinuity at \(x=3\) because the left limit approaches 3, but the right limit approaches 4, causing a break or jump in the graph.
• Removable Discontinuity: This happens when a single point is undefined on the function's graph, but the function is otherwise continuous.
• Infinite Discontinuity: This happens when the function approaches infinity at a certain point.

In the context of our exercise, we need to ensure the graph of the function shows discontinuity only at the specified point \(x=3\), with the graph breaking there to represent the jump from \(3\) to \(4\). This visual break is crucial in the understanding of discontinuous behavior in functions.

Piecewise Functions

Piecewise functions are excellent tools for defining different behaviors of a function over various intervals. They are composed of multiple functions, each applying to a different span of the domain. In our exercise, we constructed a piecewise function to handle discontinuity:

• The function is defined as \(f(x) = x\) for values less than \(x=3\).
• For \(x=3\), we set \(f(3) = 3\). This ensures the function connects smoothly from the left, keeping the function continuous from this direction.
• For values greater than \(x=3\), the function changes to \(f(x) = x+1\). This creates the needed discontinuity since it results in a different value than the left-side function, namely \(4\) when approaching from the right side.

By plotting the graph, the separate continuous lines make it evident how the function behaves differently over the different intervals, highlighting the concept of piecewise function effectively. This approach is ideal for handling situations where different rules are needed for different sections of the domain.

Limits and Continuity

The concept of limits is essential when discussing the continuity of a function. A limit helps us understand the behavior of a function as it approaches a specific point. When we say a function is continuous at \(x=3\), we mean that the left-hand limit, the value of the function at \(3\), and the right-hand limit all coincide at that point. However, if any of these values differ, a discontinuity is present.

• Left-hand limit: \( \lim\limits_{x \to 3^-} f(x) = 3 \). This is the limit as the function approaches \(x=3\) from values less than \(3\), which in our function, matches the defined value of the function at \(3\).
• Right-hand limit: \( \lim\limits_{x \to 3^+} f(x) = 4 \). This is the limit as the function approaches \(x=3\) from values greater than \(3\), which differs in our setup, creating the discontinuity.

Continuity essentially checks if the left and right limits are the same and equal to the function's value at that point. Our function is not fully continuous at \(x=3\) since the left and right limits do not match, showcasing a fundamental aspect of limits and how they relate to continuity.