The authors consider the following one-dimensional stochastic differential equation with infinite time delay and fixed jump times \(t_k\uparrow\infty\): \[ \begin{aligned} & d x(t) = x(t)\bigg\{r(t)-a(t)x(t)+b(t)x(t-\tau)+c(t)\int_{-\infty}^0x(t+\theta)d\mu(\theta) \bigg\}\\ & d t +\sigma_1(t)x(t) d W_1(t) +\sigma_2(t)x^2(t) d W_2(t)\end{aligned} \] for \( t\neq t_k\), and \[ x(t_k+)= (1+h_k) x(t_k),\;\;k\geq 1, \] where \(r, a, b, c, \sigma_i\in C_b([0,\infty))\), \(\tau\) is a positive constant stands for the finite time delay, \(\mu\) is a probability measure on \((-\infty,0]\) with finite exponential moment which gives the infinite time delay, \((W_1(t), W_2(t))\) is a standard two-dimensional Brownian motion, and \(h_k\) are constants. Sufficient conditions are presented for the distinction and the (weak) persistence of the solution. Simulations are made to illustrate the main results. It would be nice to consider the random jump time cases, for instance, by adding an additional linear jump noise to the SDE.