Here is the sketch of the function based on your request, showing the removable discontinuity at $x = 3$, the jump discontinuity at $x = 5$, and the vertical asymptote at $x = 1$.

Would you like any further details or modifications? Feel free to ask any follow-up questions!

Follow-up Questions:

1. How do we formally define and calculate the limits for removable discontinuities?
2. What are the main differences between a jump discontinuity and an infinite discontinuity?
3. How do vertical asymptotes impact the behavior of rational functions?
4. What are some other examples of functions with removable discontinuities?
5. How does the continuity of piecewise functions depend on the conditions at the boundary points?

Tip: When analyzing piecewise functions, it's helpful to graph each piece separately and then combine them. Pay special attention to the boundaries where discontinuities might occur!