# Construction of the Lipschitz function with a given Lipschitz constant and given two values

Let the function $$f\colon [a,b] \to\mathbb{C}$$ be Lipschitz and let $$|f(a)| \geq c$$ and $$|f(b)| = c$$. Is there a Lipschitz function $$g$$ such that $$|g| \geq c,$$ $$g(a)=f(a),$$ $$g(b)=f(b)$$ and Lipschitz constant of $$f-g$$ is less than epsilon for any positive epsilon?

There should be some simple counterexample.

$$\newcommand\ep\varepsilon$$Yes, it is easy to construct a counterexample here.
Indeed, if $$g_\ep$$ is such a function for each given real $$\ep>0$$ (so that $$|g_\ep| \geq c$$, $$g_\ep(a)=f(a)$$, $$g_\ep(b)=f(b)$$, and the Lipschitz constant of $$f-g_\ep$$ is less than $$\ep$$), then $$g_\ep\to f$$ pointwise (as $$\ep\downarrow0$$). So, it would follow that $$|f|\ge c$$ everywhere.
However, it is very easy to construct a Lipschitz function $$f\colon [a,b] \to\mathbb{C}$$ such that $$|f(a)|\geq c$$ and $$|f(b)| = c$$, but $$|f(x)| for some $$x\in(a,b)$$.