Let f be a bounded function on a set A, and set M = sup{f(x) : x ∈ A}, m= inf{f(x) : x ∈ A}, M = sup{|f(x)| : x ∈ A}, and m = inf{|f(x)| : x ∈ A}.(a) Show that M − m ≥ M − m. (b) Show that if f is integrable on the interval [a, b], then |f| is also integrable on this interval. (c) Provide the details for the argument that in this case we have | ) ba f| ≤) ba |f|.

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