Lipschitz classes with variable exponents \(\text{Lip}_{\alpha(t)}\) are introduced. The exponents \({\alpha(t)}\) (called test functions) are supposed to be real-valued continuous functions defined in the right neighbourhood of zero satisfying the following conditions: \[ 1)\;{\alpha(t) = \alpha + o(1)},\;\alpha\in {\mathbb R};\quad 2) \;\int_{0}^{t} \tau^{\alpha(\tau)- \beta} d\tau = \frac{\tau^{\alpha(\tau)- \beta + 1}}{\alpha + 1 - \beta} + o\left(\tau^{\alpha - \beta + 1}\right),\;\beta \alpha + 1. \] Properties of the Hilbert transform with density belonging to Lip\(_{\alpha(t)}\) are studied. Hardy-like classes of analytic functions in the unit disc with boundary functions from Lip\(_{\alpha(t)}\) are introduced and characterized.