It is proved that the necessary and sufficient condition for f (x) to be bounded in (a, b) is that f (x) has both upper and lower bounds in (a, b)

1. If f (x) is bounded in (a, b), then there exists m, always | f (x) | ≤ m, that is - M ≤ f (x) ≤ m, so f (x) has upper bound m and lower bound - M
2. If f (x) has upper bound m and lower bound n, then n ≤ f (x) ≤ m, let t = max {| m |, | n |}, then - t ≤ n ≤ f (x) ≤ m ≤ t,
That is | f (x)|

It is proved that the necessary and sufficient condition for f (x) to be bounded in (a, b) is that f (x) has both upper and lower bounds in (a, b)

It is proved that the necessary and sufficient condition for f (x) to be bounded in (a, b) is that f (x) has both upper and lower bounds in (a, b)

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