Thank you for you reply @FactChecker!FactChecker said:
Oh. The reason for not including the point ##\pi/4## is that the rest of the sentence is only about continuity on ##(-\infty, \pi/4)\cup(\pi/4, \infty)##. So there was no need to include ##\pi/4##. It wouldn't have hurt to include it.ChiralSuperfields said:
A piece-wise function is a mathematical function that is defined by different equations over different intervals. In other words, the function is divided into different pieces, and each piece is described by a different equation.
To show that a piece-wise function is continuous, you must first check that each piece of the function is continuous within its respective interval. Then, you must check that the function's value at the endpoint of each interval is equal to the limit of the function as it approaches that endpoint from both the left and right sides.
Yes, a piece-wise function can have an infinite number of pieces. This is often the case when the function has a different equation for each individual point on the number line.
Showing that a piece-wise function is continuous is important because it ensures that the function is well-defined and does not have any gaps or jumps in its graph. This is necessary for many applications in mathematics and science, where continuity is a fundamental concept.
Yes, there are specific techniques for showing piece-wise functions continuous. One common technique is using the epsilon-delta definition of continuity, where you must show that for any given epsilon value, there exists a corresponding delta value that satisfies the definition of continuity. Another technique is using the intermediate value theorem to show that the function takes on all values between two given points on the number line.