I was reading about Cauchy's theorem for existence of solution to nonlinear ODE and the book (Nhan T. Nguyen - Model-Reference Adaptive Control. A Primer) stated "continuity of f (x, t) in a closed region that contains the initial condition ensures that there exists at least one continuous solution that lies therein." As an example the following image is attached,
But isn't this ODE solvable? It can be separated and integrated to get an answer, even at zero: $$\frac{x^2}{2}-x= t +c$$
The constant can also be determined by putting Initial Conditions. I would like an example to clarify more on what Cauchy theorem states. Thanks in advance.