How to prove a function is continuous on a metric space by the theorems about continuity on a point?

Question

I'm asking for some hints or direction for my homework of proving a function is continuous on a metric space. I was given this homework after the lecture about the continuity of functions on a point in a metric space. I'm wondering what is the correct way to establish the continuity on every point in the metric space from the theorems of continuity on a single point?

Answer 1

Let $f$ be a function from a metric space $X$ to another metric space $Y$ with $d_X$ denoting the metric on $X$ and $Y$ the metric on $Y$. We say that $f$ is continuous in $p\in X$ iff for every $\varepsilon>0$ there exists a $\delta>0$ such $d_Y(f(x), f(p)) < \varepsilon$ for every $x\in X$ with $d_X(x,p)<\delta$.

Example. Let $f$ be a function from $\mathbb R$ to $\mathbb R$ given by $f(x) = x^2 + 1$. Recall that the canonical metric on $\mathbb R$ is defined by $d(x,y) = | x -y|$. To prove that $f$ is in fact continuous in every $p\in \mathbb R$ on this metric space, consider an arbitrary $p\in \mathbb R$ and let $\varepsilon>0$. Then, $$\begin{align*}d_Y(f(x),f(p)) &= \left|f(x) - f(p) \right| \\ &= \left|(x^2 + 1) - (p^2 +1)\right|\\ &= \left| x^2 - p^2\right| \\ &= \left| (x - p)(x + p)\right|\\ &= \left|(x-p)(x - p + p + p)\right|\\ &= \left|x-p\right|\left|x - p + 2p\right| \\ &\leq\left| x - p\right| (\left| x - p \right| + \left| 2p\right|) \\ &=\left|x - p\right|^2 + \vert x -p\vert\left| 2p\right| \\ & < \delta^2+\delta\left| 2p\right|,\end{align*}$$ by applying the Triangle inequality and the assumption $d_X(x,p) = |x - p|<\delta$. If there exists a $\delta>0$ such that $\delta\vert 2p\vert\leq\varepsilon/2$ and $\delta^2<\varepsilon/2$, we have proven that there exists a $\delta>0$ such that $d_Y(f(x),f(p))<\varepsilon$. That is, $f$ is continuous in $p$. Clearly, chosing $$\delta = \max\left\{\sqrt{\frac{\varepsilon}{2}},\frac{\varepsilon}{|4p|}\right\}$$ yields the desired result (if $p=0$, we can chose any $\delta<\varepsilon$).

I hope the example makes clear how to show continuity of a function.