Understanding the continuity of a function is crucial in calculus, especially when you are not provided with a graph. Discontinuities can arise in various forms, such as jumps, holes, or asymptotes, particularly in rational and piecewise functions. To identify where a function is discontinuous, we can follow a systematic approach.

For rational functions, the key to finding discontinuities lies in the denominator. A rational function is expressed as the ratio of two polynomials, and it is continuous everywhere except where the denominator equals zero. For example, consider the function:

$$f(x) = \frac{x^2 - x - 6}{x + 2}$$

Here, the denominator is \(x + 2\). Setting this equal to zero gives:

$$x + 2 = 0 \implies x = -2$$

This indicates a potential discontinuity at \(x = -2\). To determine the nature of this discontinuity, we can check if the numerator can be factored to cancel with the denominator. If it can, we have a removable discontinuity (a hole); if not, we have an asymptote. In this case, the numerator factors to \((x - 3)(x + 2)\), allowing us to cancel the \((x + 2)\) term, confirming a removable discontinuity at \(x = -2\).

Next, we examine piecewise functions, which can also exhibit discontinuities at the points where the pieces meet. For instance, consider the piecewise function:

$$f(x) = \begin{cases} 4 & \text{if } x < -1 \\ x + 1 & \text{if } x \geq -1 \end{cases}$$

To check for discontinuity, we focus on the point where the pieces connect, which is \(x = -1\). We need to evaluate the limit as \(x\) approaches \(-1\) from both sides:

1. **Left-sided limit**: As \(x\) approaches \(-1\) from the left, \(f(x) = 4\), so:

$$\lim_{x \to -1^-} f(x) = 4$$

2. **Right-sided limit**: As \(x\) approaches \(-1\) from the right, \(f(x) = x + 1\), thus:

$$\lim_{x \to -1^+} f(x) = -1 + 1 = 0$$

Since the left-sided limit (4) and the right-sided limit (0) are not equal, the overall limit does not exist at \(x = -1\). Additionally, the function value at this point is:

$$f(-1) = -1 + 1 = 0$$

Comparing the limits and the function value, we see that they do not match, confirming a discontinuity at \(x = -1\). This type of discontinuity is classified as a jump discontinuity, as the function value jumps from 4 to 0.

In summary, to determine where a function is discontinuous, analyze rational functions by setting the denominator to zero and checking for removable discontinuities through factoring. For piecewise functions, evaluate the limits at the points where the pieces meet to identify any jumps or holes. This systematic approach will enhance your understanding of function continuity and discontinuity.