Discontinuous function with bounded variation

Question

Can $\gamma:[0,1]\rightarrow\mathbb{C}$ be discontinuous and have bounded variation ?

The discontinuation is making it difficult for me to calculate the variation of any function I can think of.

Any suggestions?

Answer 1

Take for example $\gamma(t) = 0$ if $t \neq 0$, $\gamma(0) = 1$, which is discontinuous but $V_0^1(\gamma) \leq 1$.

Answer 2

If $ c :[0,1]\rightarrow \mathbb{C}$ is a curve s.t. $|c'(t)|=1$, then define $ \gamma (t)=c(f(t)) $ where $f: [0,1]\rightarrow [0,1]$ is a strictly increasing function. Then $\gamma$ has a bounded variation.

Answer 3

Let $f,g:[0,1] \to \mathbb R$ monotonic functions,which are not continuous, then $f$ and $g$ are of bounded variation. Hence $ \gamma:=f+ig$ is of bounded variation and discontinuous.