Problem
Suppose that $f(x)$ is a differentiable function and $f'(x)$ is the derivative of $f(x)$. Without aids of integral, can we prove that $f'(x)+\lambda f(x)$ has intermediate property?
Intermediate property
A (real) function $f(x)$ having intermediate property means that if $a,b\in f(\Bbb R)$ and $a
With aids of integral
Let $g(x)=f'(x)+\lambda f(x)$. For $f(x)$ is continuous, we have $f(x)$ is Riemann-integrable. Let $G(x)=f(x)+\int_0^xf(t)dt$, we have $G'(x)=g(x)$; therefore, we can apply Darboux's theorem to $G(x)$, and we've done.