Transitive Relations and functions

Question

A function by definition is a relation in which no two ordered pairs have the same first element, and every element in domain has an image in codomain. A relaton $R$ is transitive if for all values $a, b, c$: $a R b$ and $b R c$ implies $a R c$. So does it mean a function can never be transitive ?

Answer 1

Not quite, but close. The function $f:X \to X$ defined by $f(x)=x$ is a transitive relation. Your proof fails because you don't know that $b \neq c$.

Edited to add:

I believe your proof does show that $f$ is a transitive relation $\iff f \circ f = f$.