Let $A\subset \mathbb{R}$. If there exist a one-to-one map $f: [a,b]\rightarrow A$ such that $f$ is $C^1$. And we define equivalence relation on set of all C^1 function from $[a,b]$ to $A$ as two functions being equivalent if there exist diffeomorphism of $[a,b]$ on $[a,b]$ mapping $a$ to $a$ and $b$ to $b$ such that diagram we get bewteen commutes. How many different kind of equivalence classes will exist?
$\gamma_1 : [a,b]\longrightarrow A$
$\gamma_2: [a,b]\longrightarrow A$
Now $A$ to $A$, there is identity map and for equivalwnce we want map from $[a,b]$ to $[a,b]$