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In this paper, we use the techniques of fractional calculus to study the existence of a unique solution to semilinear fractional differential equation driven by a γ -Hölder continuous function 𝜃 with γ ∈ (2 3 , 1). Here, the initial condition is a function that may not be defined at zero and the involved integral with respect to 𝜃 is the extension of the Young integral [An inequality of the Hölder type, connected with Stieltjes integration, Acta Math. 67 (1936) 251–282] given by Zähle [Integration with respect to fractal functions and stochastic calculus I, Probab. Theory Related Fields 111 (1998) 333–374]. [ABSTRACT FROM AUTHOR]