Here is the graph of the floor function $f(x) = \lfloor x \rfloor$. As you can see, the function is continuous at all integer values but has a jump discontinuity at every non-integer point, fulfilling the condition you asked for.

Would you like more details about the behavior of this function or have any questions?

Here are five related questions for further exploration:

1. How does the ceiling function differ from the floor function in terms of continuity?
2. Can you describe the behavior of the fractional part function $f(x) = x - \lfloor x \rfloor$?
3. How would a graph of $f(x) = \lceil x \rceil$ (ceiling function) look like?
4. What are the real-world applications of discontinuous functions like the floor and ceiling functions?
5. How can we modify a function to make it discontinuous at specific points, say at integers instead?

Tip: Discontinuous functions like the floor function are useful in programming for tasks like rounding and discrete-event simulations.