Given $ T \in \mathbb{R}$,Find all differentiable functions that satisfy $f'(x) = f(x + T)$

Given T $\in \mathbb{R}$, Find all differentiable functions that satisfy $$f'(x) = f(x + T)$$

My Attempt. Firstly, I tried the exponential function $y=e^{ax}, T=\frac{\ln a}{a} \in (-\infty, 1/e]$ should be part of the solution, and we can extend it to the complex domain to set up a special solution $y=e^{zx}$ and solve the equation $\frac{\ln z}{z}=T$ in the complex domain, and then take $Re(e^{zx})$. My question is: Are these all the real function solutions to this equation?