The Riemann integrability of f′² + g′², f′, g′

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Suppose that, for all x in an interval [a, b], the functions f, g have finite derivatives. If, further, f′ and g′ are Riemann integrable, so are f′² and g′² and hence f′² + g′². Is it true that, conversely, the R-integrability of f′² + g′² implies that of f′ and g′? (Marcus, (2)). The answer is No, and a counter-example is given in this note. It is an elaboration of Volterra's classical construction of a derivative which is not R-integrable (3).