For the structure $\mathcal M=(\mathbb R; +,-(), \cdot, \lt, 0,1,f), n_f=1 $ show that the set of all points $x \in \mathbb R$ where $f$ is differentiable is a definable set.
My issue here is how to write a formula $\phi$ for this structure describing differentiability at a point, when the definition uses the limit definition. Do I need to describe the limit in the formula using $\epsilon, \delta$? Wouldn't that produce an extremely long formula?
Any assistance will be great.
Recall the derivative of $f$ exists at $x$ when the $\lim_{h \to 0} \frac{f(x+h)-f(x)}{h}$ exists.
The set of all such $x$ is definable by this formula: $$\exists L \forall \epsilon > 0 \exists \delta > 0 \forall h : (0 < |h| < \delta) \rightarrow (|\frac{f(x+h)-f(x)}{h} - L| < \epsilon) $$
We still are not done, because we don't have division or absolute value in the language. Can you show that these two operations are definable?