What is indicated by discontinuities in a graph?

Discontinuities in a graph typically indicate that the representation of the function does not maintain a continuous line throughout its domain. This can occur due to features such as vertical asymptotes, where the function approaches infinity as it nears a certain x-value, or holes in the graph, which represent points where the function is not defined. These characteristics illustrate that not every input leads to an output in a straightforward manner, and the graph cannot be smoothly drawn without interruption.

While it is true that a graph defined only for certain input values can showcase discontinuities, the essence of discontinuities focuses more on the inability to represent the function as a continuous line due to these specific gaps or asymptotic behavior, making the choice that emphasizes vertical asymptotes or holes the most precise indicator. The nature of discontinuities directly connects to the overall behavior of functions and how they can be graphed.