The concept of continuity in mathematics refers to a function's ability to be drawn without any breaks or gaps. In other words, the graph of a continuous function can be drawn without lifting the pen from the paper.
To prove continuity of a function, we must show that the function is defined at the point in question, the limit of the function as x approaches the point exists, and that this limit is equal to the value of the function at the point.
Yes, a function can be continuous at a single point if it meets the criteria for continuity - being defined at the point, having a limit at the point, and the limit being equal to the function's value at the point.
To prove the continuity of this function, we must show that it is defined at all points, has a limit at all points, and that this limit is equal to the function's value at each point. This can be done by breaking the function into its individual components and proving continuity for each one separately.
This function is an example of a continuous function that is not differentiable at certain points. It is commonly used in mathematics to demonstrate the concept of continuity and to challenge students to prove its continuity at various points.