All of the proofs that I have seen for the constant, identity and linear functions use the delta-epsilon definition of a limit to prove their continuity. How can this constitute a proof for continuity if it only proves the existence of a limit at $c$ but not that the function is defined at those points.
I am probably missing something, could someone explain to me why delta epsilon proofs are used rather than using the definition of continuity, $\forall \epsilon > 0\; \exists\: \delta > 0|\; |x-c| <\delta \rightarrow|f(x)-f(c)| <\epsilon$
A function being continuous in a point $c$ is equivalent to the following equality: $$ f(c) = \lim_{x \to c} f(x). $$ This means that if you wish to prove that $f$ is continuous in some point, you can simply prove that the limit exists and is equal to the value of the function.
To show this equivalence, first we have the epsilon delta definition of a limit: If the following holds, then we say $\displaystyle\lim_{x \to c} f(x) = L$. $$ \forall \varepsilon > 0,\exists \ \delta > 0 \text{ such that } 0 < |x - c | < \delta \ \Rightarrow \ |f(x) - L| < \varepsilon $$ But this is exactly your definition of continuity, just with $f(c)$ in place of $L$.