To determine where the function $g(x)$ is continuous based on the graph, we need to identify any breaks, jumps, or holes in the curve.

Analysis of the graph:

1. Discontinuities:

   • There is an open circle at $x = -2$, meaning the function is not defined at that point.
   • At $x = 0$, there is a jump discontinuity; the graph jumps from one value to another without connecting.
   • At $x = 1$, there is an open circle indicating another discontinuity.

2. Continuity:

   • The function appears continuous between:
     • $(-\infty, -2)$ (since the graph is smooth before $x = -2$),
     • $(-2, 0)$ (continuous between $x = -2$ and $x = 0$),
     • $(0, 1)$,
     • $(1, \infty)$.

Intervals of continuity:

Based on this, the function $g(x)$ is continuous on the following intervals:

• $(-\infty, -2)$
• $(-2, 0)$
• $(0, 1)$
• $(1, \infty)$

Would you like further details on continuity or have any other questions?

Related questions:

1. What types of discontinuities are there, and how do they differ?
2. How can we use limits to check for continuity at a point?
3. What is the definition of a continuous function in terms of limits?
4. How does a jump discontinuity affect the differentiability of a function?
5. How would the graph of a piecewise function impact its continuity?

Tip:

Always check for open and closed circles on a graph to determine points of discontinuity.