To determine where the function $f$ is continuous using the graph you've provided, we need to focus on the points where the graph is unbroken and where the function does not exhibit any jumps, vertical asymptotes, or holes.

Visual Inspection:

• There is an asymptote at $x = 0$ (the graph approaches infinity).
• Discontinuities are evident around $x = 1$, where the function appears to jump.

Based on these observations, the function appears to be continuous on the following intervals:

Intervals of Continuity:

Using interval notation, the function is continuous on: $(-\infty, 0) \cup (0, 1) \cup (1, \infty)$

Explanation:

• At $x = 0$, the function has a vertical asymptote, indicating a discontinuity.
• At $x = 1$, there is a jump discontinuity.

Thus, the function is continuous everywhere except at $x = 0$ and $x = 1$.

Would you like further details on continuity or an explanation of any related concepts?

Here are 5 questions for further exploration:

1. What types of discontinuities exist, and how do they differ?
2. How do you determine if a function is continuous at a specific point?
3. What is the difference between removable and non-removable discontinuities?
4. How do limits relate to the concept of continuity?
5. Can you describe how to find points of discontinuity for a rational function?

Tip: For a function to be continuous at a point $x = c$, the limit of the function as $x$ approaches $c$ from both sides must exist and be equal to the function's value at that point.