Let $X$ be a set, and let $d:X\times X\to\mathbb{R}$ be the [discrete metric](/page/Discrete%20Metric), so that $d(x,y)=0$ when $x=y$ and $d(x,y)=1$ when $x\neq y$. If $\gamma:[0,1]\to X$ is continuous from $[0,1]$ with its usual [subspace topology](/page/Subspace%20Topology) to $X$ with the metric topology induced by $d$, then there exists $x_0\in X$ such that $\gamma(t)=x_0$ for every $t\in[0,1]$.