(a) Define the average value of a real–valued function on a closed interval.
(b) Let f be continuous on [a,b]. Give a logical explanation, without performing any calculation, to show that there is at least one point c∈(a,b) at which f(c) equals the average value of f on [a,b].
(c) Construct an explicit example of a function that is not continuous on [0,1] for which no point c∈(0,1) satisfies f(c)=fˉ, where fˉ denotes the average value on [0,1]. Justify your claim.