Final answer:
To determine why a function is not continuous at a point, we need to evaluate if it's undefined, has a vertical asymptote, a jump discontinuity, or if it's just non-differentiable at that point. Non-differentiability does not imply discontinuity, but the other conditions suggest breaks in the function's graph.
Step-by-step explanation:
The student is asking about the continuity of a function at a certain point. Continuity at a point requires the function to be defined at that point, the limit of the function as it approaches the point to exist, and for the limit to equal the function's value at that point.
The conditions given suggest different types of discontinuity. For example, if a function has a vertical asymptote at a point, it means the values of the function go to infinity as the function approaches that point, indicating the limit does not exist.
A jump discontinuity occurs when the function has two different limiting values from the left and the right side of the point. Being non-differentiable refers to the function not having a derivative at the point, but this does not necessarily imply discontinuity. Lastly, if the function is not defined at a point, then it cannot be continuous there because continuity requires the function to have a value at that point.