Final answer:
A function is discontinuous at a point a if it does not meet the criteria for continuity, which include the function being defined at a, the limit as x approaches a existing, and the function's value at a matching the limit. Reasons for discontinuity can include jump, infinite, or point discontinuities.
Step-by-step explanation:
A function is discontinuous at a given number a when it does not satisfy the criteria for continuity at that point. For a function to be continuous at a point a, three conditions must be met: the function must be defined at a, the limit as x approaches a must exist, and the value of the function at a must equal that limit.
Several reasons can cause a function to be discontinuous at a number a. One possibility is a jump discontinuity where the left-hand and right-hand limits do not match. Another is an infinite discontinuity where the function approaches infinity as x gets closer to a. A third reason might be a point discontinuity, also known as a removable discontinuity, where the function is defined at a, but its value does not match the limit.
In a case where the function is double-valued or diverges and is not normalizable, it is clear that the function is not continuous since it fails to meet the aforementioned criteria for continuity. Additionally, if a function is restricted to a certain domain, its continuity is only considered within that domain, and it cannot be discontinuous where it is not defined.