Problem 30 - Calculus of a Single Variable

Question

Continuity of a Function In Exercises \(27-30,\) discuss the continuity of each function. $$ f(x)=\left\\{\begin{array}{ll}{x,} & {x<1} \\ {2,} & {x=1} \\ {2 x-1,} & {x>1}\end{array}\right. $$

Step-by-Step Solution

Answer

The function \( f(x) \) is continuous at all points except \( x = 1 \).

Step 1: Analyze the expression at \( x<1 \)

When \( x<1 \), the function is defined as \( f(x) = x \). This is a simple linear function, which is continuous for all x, including \( x = 1 \).

Step 2: Analyze the expression at \( x = 1 \)

At \( x = 1 \), the function is defined as \( f(x) = 2 \). The value of the function is defined explicitly at this point, so there is no issue of continuity here.

Step 3: Analyze the expression at \( x>1 \)

When \( x>1 \), the function is defined as \( f(x) = 2x - 1 \). This is also a linear function which is continuous for all x. However, notice that at \( x = 1 \), this expression gives \( 2(1)-1 = 1 \), which does not match the value of the function at \( x = 1 \). So there is a discontinuity at \( x = 1 \).

Step 4: Conclude

The function \( f(x) \) is continuous for all \( x \) except at \( x = 1 \), where the value of the function does not match the limit of \( f(x) \) as \( x \) approaches 1 from the right.

Key Concepts

Piecewise functions

Piecewise functions are quite unique. They are defined using different expressions for different intervals of the domain. This means a piecewise function can behave differently depending on the value of the input, or where you are along the x-axis.
When analyzing a piecewise function:

• Identify the segments and the corresponding expressions.
• Look at the boundaries between segments.
• Check how the function transitions from one segment to another.

In the given exercise, the function has three pieces:

• For values less than 1, it's linear with the equation of a line.
• At the point where x equals 1, it suddenly becomes a constant value.
• For values greater than 1, it again takes the form of a linear equation.

It's how these individual pieces work together that affects whether the function is continuous everywhere or has points of discontinuity.

Limits

Limits are crucial in determining the behavior of functions at specific points. The limit describes what a function approaches as the input gets closer to a specific point along the x-axis.
To evaluate limits for piecewise functions, especially at the transition points between pieces:

• Determine the left-hand limit (as you approach the point from the left).
• Determine the right-hand limit (as you approach the point from the right).

For a function to be continuous at a point, the left-hand limit and the right-hand limit must be the same and must equal the value of the function at that point.
In the exercise, at x = 1, the left-hand limit is 1 (since f(x)=x for x<1), and the right-hand limit is also 1 (from f(x)=2x-1 for x>1). However, the function value at x = 1 is 2, leading to a discontinuity.

Discontinuous points

Discontinuous points occur where a function "jumps," where the transition from one segment of a piecewise function to the next is not smooth.
In simpler terms, if you were drawing the graph of a function, a discontinuity is where you would need to lift your pencil off the paper to continue drawing the next part.
In the exercise function, at x = 1 is a point of discontinuity:

• The value of the function jumps from what the limit suggests it should be (1) to what it actually is defined to be (2).
• This creates a "gap" or a "jump" in the function's path on the graph.

By identifying discontinuous points, we better understand where a function falls short of being smooth or seamless across its domain.

Linear functions

Linear functions are among the simplest types of functions, represented by the equation of a straight line, usually in the form \( y = mx + b \). Here, \( m \) is the slope, indicating the steepness of the line, and \( b \) is the y-intercept.

• They are continuous everywhere on their domain. There's no break or jump at any point along the line.
• Smooth and predictable, they make analyzing the continuity of piecewise functions easier.

In the exercise:

• The segment of the function defined as \( f(x) = x \) for \( x < 1 \) is linear.
• The expression \( f(x) = 2x - 1 \) for \( x > 1 \) is also linear.

These linear segments are continuous independently, but the transition between them and the constant value at \( x = 1 \) is what leads to discontinuity. Understanding the behavior of these segments helps identify where the smooth flow of the function may be interrupted.