Question

In Exercises \(11-18,\) use the function \(f\) defined and graphed below to answer the questions. $$f(x)=\left\\{\begin{array}{ll}{x^{2}-1,} & {-1 \leq x<0} \\\ {2 x,} & {0 < x < 1} \\ {1,} & {x=1} \\ {-2 x+4,} & {1 < x < 2} \\ {0,} & {2 < x < 3}\end{array}\right.$$ Writing to Learn Is it possible to extend \(f\) to be continuous at \(x=0 ?\) If so, what value should the extended function have there? If not, why not?

Short answer

No, the function cannot be extended to be continuous at x=0, because the right-hand limit and left-hand limit at x=0 do not match.

Step-by-step solution

Calculate Left-hand limit as x->0

The left-hand limit as x->0 uses the function \(x^{2}-1\). Substituting x=0 into the equation results in \((0)^{2}-1 = -1\).

Calculate Right-hand limit as x->0

The right-hand limit as x->0 uses the function \(2x\). Substituting x=0 into this equation results in \(2*0 = 0\).

Compare values to test for continuity

As per the definition of continuity at a point, the left-hand limit, right-hand limit and value of the function at that point should match. Looking at the left-hand limit (-1), the right-hand limit (0) and the value at x=0 (not given), we see that the limits are not the same therefore the function is not continuous at x=0.

Key concepts

Limit of a Function

The concept of the limit of a function is foundational to calculus. It describes what value the function approaches as the variable gets closer to a particular point. A limit does not always equal the function's value at that point, which is why it's so important for understanding the behavior of functions, especially those that are not straightforward.

When discussing limits, it's crucial to differentiate between the 'actual value at a point' and the 'approaching value', as they may not be the same. Limits are commonly used to define the very concepts of continuity, integrals, and derivatives, which are the cornerstones of calculus. By understanding limits, one can grasp why a function behaves a certain way near a point, even if the function isn't actually defined at that very point.

Piecewise-defined Functions

Piecewise-defined functions are those that have different expressions or rules for different intervals of their domain. They're essential in mathematical modeling because they let us describe complex behaviors that change over different conditions or ranges. For a piecewise-defined function like the one in the exercise, you have multiple 'pieces' of functions each defined on specific intervals.

When evaluating these functions at certain points, especially at the boundaries of these intervals, you need to determine which 'piece' of the function applies. This can complicate finding limits and testing for continuity, which is why clarity in understanding each 'piece' and the rules for its interval is important.

Testing for Continuity

Testing for continuity at a point involves checking three conditions: the function must be defined at the point, and the limit of the function as it approaches the point from both the left and the right must exist and be equal to each other and to the actual value of the function at that point.

For piecewise functions, this can be tricky since you are dealing with multiple 'pieces' that may have different behaviors near the point in question. The steps provided in the solution neatly demonstrate these conditions. In the exercise's context, since the left-hand and the right-hand limits at x=0 are not equal, the function fails the test for continuity at x=0.

Left-hand and Right-hand Limits

The left-hand limit of a function at a point is the value that the function approaches as the variable approaches the point from the left (from smaller values). Similarly, the right-hand limit is the value the function approaches as the variable approaches the point from the right (from larger values).

To be continuous at a point, a function’s left-hand and right-hand limits must coincide. This is a crucial check for continuity, especially for piecewise functions where the 'direction' from which you approach a point could change the 'piece' of the function you are examining. As noted in the exercise’s steps, the difference in these limits at x=0 indicates a discontinuity in the function at that point.