# real analysis – Does the existence of unilateral boundaries in a delimited domain imply the delimitation of function?

I read a proof of the following lemma:

Lemma: Let $$f: (a, b)$$

to mathbb {R} \$ a non-negative function such as each restriction $$f | _ {[a+epsilon,b]}$$ is integrable for all $$epsilon> 0$$. Then the integral improper $$int_a ^ bf: = lim _ { epsilon to0 ^ +} int_ {a + epsilon} ^ bf (x) dx$$ converges if, and only if, there is a $$K > 0$$ such as $$int_ {a + epsilon} ^ bf (x) dx the K$$ for everyone $$epsilon$$.

Proof: Define $$phi ( epsilon) = int_ {a + epsilon} ^ bf (x) dx$$. Because $$f 0$$it follows that $$phi$$ defined for $$0 < epsilon is monotonous not growing. Hence the unilateral limit of $$phi$$ as $$epsilon to 0 ^ +$$ exists if and only if $$phi$$ is delimited. Then the lemma follows.

I am aware that the limitation and monotony of the function imply the existence of unilateral limits. But I do not see how to justify the opposite by using the hypothesis. If anyone could explain me further, this would be a big help. Thank you.