As you know by the eplsilon-delta definition of the limit,
$$0<{|x-c|}<δ$$
So $x$ cannot be $c$ in
$$f'(x)= \lim_{x \rightarrow c} \frac{f(x)-f(c)}{x-c} \dots (*)$$
So $\Delta x$ also cannot be $0$.
Then how to plug in zero for $\Delta x$ when evaluating $(*)$ with $\Delta x $ difference?
Like:
f(x)=x^2,f'(1) lim Δx->0 f(1+Δx)-f(1)/Δx lim Δx->0 (1+Δx)^2-1/Δx lim Δx->0 Δx^2+2Δx/Δx lim Δx->0 Δx+2
and then just plug in $0$ to make the value $2$?
Can we prove that $\Delta x$ is continuous?