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In this paper we study the Hardy–Littlewood maximal operator in variable exponent spaces when the exponent is not assumed to be bounded away from 1 and∞. Within the framework of Orlicz–Musielak spaces, we characterize the function space X with the property that Mf∈ Lp (·) if and only if f∈ X, under the assumptions that p is log-Hölder continuous and

1⩽ p−⩽ p+⩽∞.