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When we talk about a removable discontinuity, we're referring to a point on a function's graph where there's a hole. This happens when we can "fix" the function at that point by redefining it. It's like a small interruption that can easily be patched up.

Take, for instance, the function \( f(x) = \frac{x^2 - 4}{x^3 - 8} \). We found that at \( x = 2 \), the function is initially discontinuous. However, since we managed to factor and cancel \( (x-2) \) in the numerator and denominator, the discontinuity at this point was removable. We are left with the continuous function \( \frac{x+2}{x^2+2x+4} \) at \( x = 2 \).

In this case, the removable discontinuity means we "cancel out the problem", so the function continues smoothly as if the interruption never existed. This is a key feature of removable discontinuities—it's all about simple adjustments.

Non-removable discontinuities are a bit different. These occur where no simple tweaking can re-establish the continuity of the function. They often arise in step or jump form.

Let's look at a piecewise function like \( f(x)= \begin{cases} 2x - 3, & x \leq 2 \ x^2, & x > 2 \end{cases} \). Here, there's a break at \( x = 2 \). The limit from the left side is 1, whereas from the right it is 4. Different limits mean a definite gap; thus, this is a non-removable discontinuity. No matter how we redefine \( f(2) \), we can't fill this gap seamlessly.

Another example is the function \( f(x)= \begin{cases} 3x^2 + 5, & x eq 1 \ 6, & x = 1 \end{cases} \). At \( x=1 \), the graph doesn't smoothly continue, because the function value (6) and the limit (8) are not the same. This separation confirms it's a non-removable discontinuity.

Piecewise functions can be thought of as mathematical patchwork quilts. They stitch different function rules over specified intervals.

In a piecewise function, each "piece" functions according to its own rule in its domain. For example, consider the function \( f(x)= \begin{cases} 2x - 3, & x \leq 2 \ x^2, & x > 2 \end{cases} \). This is composed of two parts, a linear segment when \( x \leq 2 \), and a quadratic segment when \( x > 2 \). The potential for discontinuity arises at the juncture where these segments meet, in this case, \( x = 2 \).

Another example, \( f(x)= \begin{cases} 3x^2 + 5, & x eq 1 \ 6, & x = 1 \end{cases} \), comprises a continuous quadratic expression applied universally except at a specific point \( x = 1 \).

These functions can present both removable and non-removable discontinuities, making it essential to evaluate limits and endpoints to fully understand their behavior across specified intervals.