I am a first reader of Differential Equation. I was solving a differential equation whose solution is $|f(x)|= c$. Now my question is can I write that the solution is $f(x)= k$. If $f(x)$ is continuous then I can remove the mod.
But I am not sure whether the function $f(x)$ is continuous or not. (Here $c$ and $k$ are constants)
So that's why my question is the following. Is the solution of every differential equation CONTINUOUS in it's given domain?
Can anyone please help me to understand this?
As you know a differentiable function in continuous.
A solution to a differential equation $y'= f(x,y)$ is a function $y=y(x)$ which satisfy $$ y'(x) = f(x,y(x))$$
That is the solution $y=y(x)$ is necessarily differentiable and its derivative satisfies $$ y'(x) = f(x,y(x))$$
In your example of |f(x)|=c, you need an initial condition to make a decision about f(x)=c or f(x)=-c but in either case your function should be continuous on the given domain.