Question

Sketch the graph of a function that has domain \([0,6]\) and is continuous on \([0,2]\) and \((2,6]\) but is not continuous on \([0,6]\).

Short answer

The graph has a jump discontinuity at x=2.

Step-by-step solution

Identify the Domain and Continuity Requirements

The domain of the function is given as [0, 6]. The function is continuous on [0, 2] and (2, 6] but not on [0, 6]. This implies a discontinuity at the point x=2.

Choose the Type of Discontinuity

For the discontinuity at x=2, we can select a simple type of discontinuity such as a jump discontinuity. This means that the limit from the left as x approaches 2 doesn't equal the function value at x=2 or the limit from the right.

Design a Continuous Function on [0,2]

Sketch a simple continuous function on [0, 2], such as f(x) = x^2. It starts at (0,0) and ends at (2,4).

Introduce the Discontinuity at x = 2

At x=2, define that the limit of f(x) as x approaches from the left is not equal to f(2). For example, let f(2) = 5, which creates a jump discontinuity at x = 2.

Design a Continuous Function on (2,6]

For the interval (2, 6], maintain continuity by choosing a function like f(x)=2. Ensure it starts at (2, 5) (open circle) and ends at (6, 2).

Sketch the Graph

Draw the function based on the designed steps: a curve from (0,0) to (2,4), a jump to (2,5), and then a straight line parallel to the x-axis from (2,5) to (6,2).

Key concepts

Continuous Function

A continuous function is one that does not contain any breaks, jumps, or holes, within the intervals where it is defined. Essentially, you can draw a continuous function without lifting your pen off the paper. Mathematically, a function \( f(x) \) is continuous at a point \( x = c \) if the limit of \( f(x) \) as \( x \) approaches \( c \) from both sides equals the function value at that point: \[ \lim_{{x \to c^-}} f(x) = f(c) = \lim_{{x \to c^+}} f(x) \] For the exercise given, the function must be continuous on the intervals \([0, 2]\) and \((2, 6]\), but is not continuous across the entire domain \([0, 6]\). This means there is a point on the graph, specifically \(x = 2\), where this continuity is interrupted.

Jump Discontinuity

Jump discontinuity occurs at a point where the left-hand limit and right-hand limit of a function exist but are not equal to each other. It's a type of discontinuity where the graph of the function literally "jumps" from one position to another. In the context of our example, a jump discontinuity is introduced at \(x = 2\). To illustrate this jump, you can consider a scenario where as \(x\) approaches 2 from the left, the function values converge towards 4, but at \(x = 2\) itself, the function suddenly jumps to a value of 5. This is why we see the graph moving abruptly from one point to another without connecting the two points.

Domain of a Function

The domain of a function refers to the complete set of possible input values (\(x\) values) for which the function is defined. In simpler terms, it is the range of "experiment" over which the function produces an output. For our example, the domain is given as \([0, 6]\), which means the function is defined for every \(x\) value from 0 up to and including 6. However, it's vital to note that while the function is defined over this entire range, it isn't continuous over it due to a discontinuity point at \(x = 2\). This tells us we can have functions that are discontinuous on their domains, specifically handled by defining conditions at certain points.

Graph Sketching

Graph sketching is a visual representation of a function's behavior over a specified domain. It involves plotting points to show how the function behaves across its intervals, where it is continuous, and where any discontinuities occur. Following the step-by-step solution, this process begins by sketching a simple continuous curve on \([0, 2]\), such as \(f(x) = x^2\). This segment of the graph is straightforward and smoothly connects points from (0,0) to (2,4). Next, introduce a jump discontinuity at \(x=2\), depicting the graph jumping vertically to a new point, say (2,5). Finally, draw another segment along \((2, 6]\), showing the function continues monotonously after the discontinuity with something like a horizontal line representing \(f(x) = 2\). This way, the graph clearly exhibits both continuous sections and the noteworthy jump discontinuity.