Continuity definition confusion I am a little confused about the definition of continuity;
A function $f : E \to \mathbb R$ is continuous at $c$ if for every $\epsilon > 0$ there exists a $\delta > 0$ such that if $|x − c| < \delta$ and $x \in E$ then $|f(x) − f(c)| < \epsilon.$
First of all, does 'for every $\epsilon > 0$ there exists a $\delta > 0$' mean that delta depends on epsilon as I thought the definition suggests that epsilon depends on delta. Also, how does this definition show continuity, if anyone can explain it in plain English. If I look at the definition, I have imagine the graph where $|x − c|$ is within delta and which dictates the value of epsilon for which $|f(x) − f(c)|$ is within epsilon but I don't have any idea of how it relates to the informal definition of continuity - where you can draw the graph without taking your pen off the paper.
 A: In answer to the first question, yes, it means you are able to express $\delta$ as a function of $\epsilon$. 
As for the plain English: suppose someone gives you a graph, with some function $f(x)$ plotted on it, and they draw a strip going between two $y$ values, centred around $x=c$. The width of the strip is $2\delta$. You'll be left with something that looks like this 

Then someone asks you "Could you tell me if the function around the point $c$ (in the picture, I have chosen without loss of generality $c=-3$ as an example) stays inside this band, no matter how large or small I make it?". To answer this question, we make an observation, that no matter how big the band is, we can always define a range of the function in the neighbourhood of the point. This corresponds to the following picture:

Now, in answer to the question, is this function continuous at the point $c$, we must be able to shrink the width of the orange strip as small as we want around $c$, making sure that the point $f(c)$ stays inside the blue strip. Continuity is telling you the function at the point $c$ should tend to $f(c)$ as you approach from some different $x \neq c$. In the graph we recognise that the strip shrinking is analogous to us approaching $x=c$ from some other $x\neq c$ in the neighbourhood bound by the strip. We see this visually too

Summed up, continuity at a point $c$ is answering the question "Does the function around $c$ tend to $f(c)$ around a local neighbourhood?"
What would a function discontinuous at $c$ look like? Well, the point is, the function musn't be able to tend to $f(c)$ from the local neighbourhood. In our example, let's define a new function
$$g(x) = \left\{\begin{matrix} f(x) & x\neq c \\ f(c) + 8 & x=c \end{matrix}\right.
$$
In the picture this corresponds to one point shifted up (i.e. the red blob).
You see now that even if we start the procedure again with a large enough neighbourhood

As we start to shrink down again (equivalent to taking the limit $x\longrightarrow c$), the point $g(c)$ no longer lies inside our blue rectangle: the function is no longer continuous at $c$!

