An example of a non-integrable function f is f(x) = 1/x on the interval [0, 1]. This function is not integrable because it has an infinite discontinuity at x = 0.
Functions can have infinite discontinuities at certain points if they approach infinity or negative infinity at that point. In the case of f(x) = 1/x, as x approaches 0 from the positive side, the function approaches positive infinity. Similarly, as x approaches 0 from the negative side, the function approaches negative infinity.
Despite f(x) not being integrable on its own, f^2(x) is integrable because the function becomes continuous at x = 0. This is because the negative and positive infinities cancel each other out when squared. Similarly, |f(x)| is integrable because it is essentially the absolute value of f(x) and the negative and positive infinities also cancel out.
Yes, a function can be non-integrable at a specific point but still be integrable on a larger interval. This is because the integration process involves taking the limit of the function as the interval approaches the point in question. If the limit does not exist or is infinite, then the function is not integrable at that point. However, the function may still be integrable on a larger interval as the limit may exist and be finite within that interval.
Yes, there are other examples of non-integrable functions where |f| and f^2 are integrable. One example is f(x) = sin(1/x) on the interval [0, 1]. This function has an infinite discontinuity at x = 0, making it non-integrable. However, |f(x)| = |sin(1/x)| is integrable on this interval, as is f^2(x) = sin^2(1/x).