Let f and g be functions defined on (possibly different) closed intervals, and assume the range of f is contained in the domain of g so that the composition g ◦ f is properly defined. (a) Show, by example, that it is not the case that if f and g are integrable, then g ◦ f is integrable. Now decide on the validity of each of the following conjectures, supplying a proof or  ounterexample as appropriate. (b) If f is increasing and g is integrable, then g ◦ f is integrable. (c) If f is integrable and g is increasing, then g ◦ f is integrable.

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A: f is surjective.

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Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.