Let $f:X\rightarrow Y$ be a continuous function between two metric space then the collection $\mathfrak{A}=\{f \}$ is equicontinuous if and only if... I found the question in a book named Topology of Metric spaces. Question asked for finding iff condition for the given family to be equicontinuous.
My answer: $\mathfrak{A}$ is equicontinuous if and only if $f$ is continuous. Actually I am confused about the question. In the question it is given that $f$ is continuous. Please someone help me clarifying my doubts.
I assume they are just asking you to give the definition of equicontinuity. A collection $\mathcal{U}$ is equicontinuous if and only if for all $\epsilon>0$, there exists a $\delta>0$ so that whenever $d_X(t,s) < \delta$, $\sup_{f\in \mathcal{U}}d_Y(f(t),f(s)) < \epsilon$. In other words whenever $d_X(t,s) < \delta$, $d_Y(f(t),f(s)) < \epsilon$ for all $f\in \mathcal{U}$. Notice this is stronger than continuity. Can you see why?