Definition of $\mathcal C^0([a,b])$ and $\mathcal C^1([a,b])$ May question may be obvious, an sorry for that. In my lecture, we say that $f\in \mathcal C^0([a,b])$ if $f\in \mathcal C^0(a,b)$ and $\lim_{x\to a^+}f(x)$ and $\lim_{x\to b^-}f(x)$ exist. 
So, for me $$\mathcal C^0([a,b])=\{f:[a,b]\to \mathbb R\mid f\text{ continuous}\}.$$
So, since for a function $f:[a,b]\to \mathbb R,$ and $c\in [a,b]$,
$\lim_{x\to c}f(x)=\ell$ means that $$\forall \varepsilon >0, \exists \delta >0:\forall x\in [a,b], |x-c|<\delta \implies |f(x)-\ell|<\varepsilon ,\tag{*}$$
Q1) I don't really see the necessity to precise $\lim_{x\to a^+}f(x)$ exist. Indeed, ix $x<a$, then $x\notin [a,b]$, so $(*)$ still hold. Therefore, $\lim_{x\to a}f(x)=f(a)$ sounds to be a good notation, no ?
Q2) Same for $\mathcal C^1([a,b])$ it's the set of function s.t. $f\in \mathcal C^0([a,b])\cap \mathcal C^1(a,b)$ s.t. $\lim_{x\to a^+}f'(x)$ and $\lim_{x\to b^-}f'(x)$ exist. Is it really necessary to precise $x\to a^+$ and $x\to b^-$ because on $[a,b]$, for example $\lim_{x\to a^+}f'(x)=\lim_{x\to a}f'(x)$ (according to the definition $(*)$).  
 A: The two sets of functions
$$
\mathcal C^0([a,b]) = \left\{\, f\colon (a,b)\to\mathbb R \,\middle|\, \text{$f$ is continuous and $\lim_{x\to a^+} f(x)$, $\lim_{x\to b^-} f(x)$ exist} \,\right\}
$$
and
$$
\widetilde{\mathcal C^0}([a,b]) = \left\{\, f\colon [a,b]\to\mathbb R \,\middle|\, \text{$f$ is continuous} \,\right\}
$$
are naturally in bijection: Every $f\in\mathcal C^0([a,b])$ uniquely extends to a continuous function on $[a,b]$ and every $f\in\widetilde{\mathcal C^0}([a,b])$ has restriction $f|_{(a,b)}\in \mathcal C^0([a,b])$. So in this way, the two definitions are equivalent. However, in the first definition you can't just write $f(a)$ or $f(b)$ since the domain is $(a,b)$.
For your definition of $\mathcal C^1([a,b])$ it is indeed irrelevant wether you write $\lim_{x\to a^+} f'(x)$ or just $\lim_{x\to a} f'(x)$ since the domain of $f$ is just $(a,b)$ anyway.
However, writing it this way might be benefical when you start with some $g\colon \mathbb R\to\mathbb R$ differentiable on $(a,b)$ and want to know if $f=g|_{(a,b)}$ is in $\mathcal C^1([a,b])$. In this setting
$$
\lim_{x\to a} f'(x) = \lim_{x\to a^+} f'(x) = \lim_{x\to a^+} g'(x)
$$
but $\lim_{x\to a} g'(x)$ might not exist even when $\lim_{x\to a^+} g'(x)$ does exist.
