In order to prove continuity of a function, you need to show that
the left side limit and right side limit of the function are identical.
Then you need to show that the function value and (left-right) limit are same... You do this all on the normal function.

Now my question, would it be allowed to use the first derivation of the function instead of the normal function?

Example (using normal function): Prove that $f(x) = x^{2}$ is continuous at $x\_{0}=1$.

$\lim\_{x\rightarrow1^{-}}x^{2}=1$

$\lim\_{x\rightarrow1^{+}}x^{2}=1$

Thus the function is continuous.

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Example (using first derivative): Prove that $f(x) = x^{2}$ is continuous at $x\_{0}=1$.

$f'(x)=2x$

$\lim\_{x\rightarrow1^{-}}2x=2$

$\lim\_{x\rightarrow1^{+}}2x=2$

Thus the function is continuous.