Let f be a monotonically increasing function from [0, 1] into [0, 1]. Which of the following statements is/are true? (a)f must be continuous at all but finitely many points in [0, 1] (b)f must be continuous at all but countably many points in [0, 1]( c) f must be Riemann integrable (d)None of these?

Question

Answer

Monotonically Increasing Function

A monotonically increasing function is a function that always increases or stays the same as its input increases. In other words, for any two points x and y in the domain of the function, if x ≤ y, then f(x) ≤ f(y).

Statement (a): f must be continuous at all but finitely many points in [0, 1]

This statement is true. Since f is monotonically increasing, it can only have a finite number of discontinuities in the interval [0, 1]. This is because if f has an infinite number of discontinuities, it would violate the monotonicity property.

To prove this, let's assume that f has infinitely many discontinuities in [0, 1]. Since f is increasing, this means there would be infinitely many points where f jumps from a smaller value to a larger value. However, this contradicts the definition of a monotonically increasing function.

Therefore, f must be continuous at all but finitely many points in [0, 1].

Statement (b): f must be continuous at all but countably many points in [0, 1]

This statement is also true. Any function with finitely many discontinuities must be continuous at all but countably many points. This is because the set of all discontinuities is countable.

A countable set is a set that can be put into one-to-one correspondence with the set of natural numbers (or a subset of the natural numbers). In other words, the elements of a countable set can be listed or enumerated.

Since the set of discontinuities of f is finite, it can be listed or enumerated, making it countable. Therefore, f must be continuous at all but countably many points in [0, 1].

Statement (c): f must be Riemann integrable

This statement is not necessarily true. While it is true that any monotonically increasing function on a closed and bounded interval is Riemann integrable, the given information does not specify that f is bounded. Without the boundedness assumption, we cannot conclude that f is necessarily Riemann integrable.

Therefore, none of the given statements is false.

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