proving continuity using epsilon delta

Question

There are many excersises about proving continuity using epsilon delta definition, but there are many varitations. Maybe the problem lies in me, not understanding the concept of continuity very well.
What are differences if I am asked:

1. to prove that some function is continuos (e.g. $f:\mathbb{R}\rightarrow\mathbb{R}, x\mapsto |-2x+3|$)
2. to prove that some function is continuos at a given $x$, which is element of the domain
3. to check if a function is continuos at all $x$'s of the domain (e.g. $g:\mathbb{R}\rightarrow\mathbb{R}, x\mapsto 2x^4-2$)

is 1 and 3 the same? I am pretty confused about it all.
Thanks in advance.

Answer 1

Indeed, 1 and 3 are the same. The problem 2 is different. For instance, if you have the function $f\colon\mathbb{R}\longrightarrow\mathbb R$ define by$$f(x)=\begin{cases}1&\text{ if }x\geqslant0\\0&\text{ otherwise,}\end{cases}$$you cannot possibly prove that it is continuous (since it isn't). But it is a perfectly reasonable question to ask you to prove that it is continuous at $2$ or that it is discontinuous at $0$.

Answer 2

Such definitions in analysis can be written using quantifiers. One need only understand what $\forall$ and $\exists$ mean and then the rest of the definitions should be straightforward. I’ve written some of them below.

1. A sequence $a_n$ converges to a limit $\ell:$ $$\forall \epsilon>0\quad\exists N\quad\forall n>N\quad |a_n-\ell|<\epsilon$$
2. A function $f$ is continuous at a point $x$: $$\forall \epsilon>0\quad\exists\delta>0\quad\forall y\quad|x-y|<\delta\Rightarrow|f(x)-f(y)|<\epsilon$$
3. A function $f$ is continuous (i.e. continuous at all points): $$\forall x\quad\forall \epsilon>0\quad\exists\delta>0\quad\forall y\quad|x-y|<\delta\Rightarrow|f(x)-f(y)|<\epsilon$$
4. A function $f$ is uniformly continuous: $$\forall \epsilon>0\quad\exists\delta>0\quad\forall x,y\quad|x-y|<\delta\Rightarrow|f(x)-f(y)|<\epsilon$$

One needs to

1. Understand what $\forall$, $\exists$, and $\Rightarrow$ mean.
2. Understand how to negate such propositions
3. Understand how to convert such propositions into the structure of a proof.

Step three goes as follows:

• read from left to right
• replace $\forall a\in A$ with “Let $a\in A$” (sometimes the set $A$ is implicit—usually $\Bbb R$ or the domain of $f$)
• replace $\exists x$ with a construction for what $x$ should be (and if you saw e.g. $\exists x>0$ then you need a proof that $x>0$ unless it’s obvious)
• replace $P\Rightarrow Q$ with “Suppose $P$” and then a proof of $Q$