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Quiz 1
Consider the metric spaces $(M,d_1)$ and $(N,d_2)$ where $M=N=\mathbb{R}$ and $d_1(x,y)=d_2(x,y)=\left\{ \begin{array}{ll} 0, & \quad x=y \\ 1, & \quad x \neq y. \end{array} \right.$
Prove that any function $f \colon M \rightarrow N$ is continuous.
Proof: Let $\epsilon > 0$ and let $p \in M$. Choose $\delta=\dfrac{1}{2}$. Then, due to the definition of the metric, any $q \in M$ that satisfies $d(p,q)<\delta$ must actually be equal to $p$, i.e. $q=p$ is the only option available for this $\delta$ (in fact, for any particular $0<\delta<1$). Therefore $q=p$ and we compute $$d_1(f(p),f(q))=d_1(f(p),f(p))=0<\epsilon,$$ completing the proof. $\blacksquare$