Question

Give the three conditions that must be satisfied by a function to be continuous at a point.

Short answer

Answer: For a function to be continuous at a point, three conditions must be met: 1. The limit of the function exists at that point. 2. The function is defined at that point. 3. The limit of the function is equal to the function value at that point.

Step-by-step solution

Condition 1: Existence of limit at the given point

To be continuous at a given point, the function must have a limit at that point. This means that as the input (or x-values) approaches the point in question, the output (or y-values) of the function must approach a particular value, which can be finite.

Condition 2: Existence of function value at the given point

The function must be defined at the given point, meaning there must be a specific output (y-value) for the input (x-value) at that point.

Condition 3: Equality of limit and function value at the given point

Lastly, for a continuous function, the limit and function value at the given point should be equal. This means that the y-value that the function approaches as x approaches the point is the same as the actual output value of the function at that point. Therefore, a function is continuous at a point if the above three conditions are satisfied, i.e., the limit exists at that point, the function is defined at that point, and the limit is equal to the function value at that point.