For functions in the Lebesgue space L(ℝ+), a modified strong dyadic integral J α and a modified strong dyadic derivative D (α) of fractional order α > 0 are introduced. For a given function f ∈ L(ℝ+), criteria for the existence of these integrals and derivatives are obtained. A countable set of eigenfunctions for the operators J α and D (α) is indicated. The formulas D (α)(J α(f)) = f and J α(D (α)(f)) = f are proved for each α > 0 under the condition that $$\smallint _{\mathbb{R}_ + } f(x)\;dx = 0$$ . We prove that the linear operator $$J_\alpha :L_{J_\alpha } \to L\left( {\mathbb{Z}\mathbb{R}_ + } \right)$$ is unbounded, where $$L_{J_\alpha }$$ is the natural domain of J α. A similar statement for the operator $$D^{(\alpha )} :L_{D^{(\alpha )} } \to L\left( {\mathbb{R}_ + } \right)$$ is proved. A modified dyadic derivative d (α)(f)(x) and a modified dyadic integral j α(f)(x) are also defined for a function f ∈ L(ℝ+) and a given point x ∈ ℝ+. The formulas d (α)(J α(f))(x) = f(x) and j α(D (α)(f)) = f(x) are shown to be valid at each dyadic Lebesgue point x ∈ ℝ+ of f.