If $\dfrac{\partial N}{\partial x} - \dfrac{\partial M}{\partial y} > 0$ at a point $P$ then how can we say $\dfrac{\partial N}{\partial x} - \dfrac{\partial M}{\partial y} > 0$ in some region surrounding $P$.
text books says - continuity of derivatives. I just know that if F is differentiable then it is continuous. I did not understand how $> 0$ at Point $P$ translates to $> 0$ in some region around $P$
Let $f : \mathbb R ^n \to \mathbb R$ be some continuous function. Suppose we know $f(x) > 0$ for some $x \in \mathbb R ^n$. By continuity (choosing $\varepsilon = \frac{f(x)}{2}$) there exists a $\delta > 0$ such that for all $y$ with $\lVert x - y \rVert < \delta$ we have $\lvert f(x) - f(y) \rvert < \delta$. Thus for these $y$ we have $f(y) > \varepsilon = \frac {f(x)}{2} > 0$ (by the triangle inequality). Thus if $f(x) >0$ we also have that $f$ is greater than $0$ on a region surrounding $x$.
Now take $f = \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} $. Then $f$ is continuous (if $N,M \in \mathcal C ^1$) and $f(P) > 0 $. The above reasoning yields that $f$ is greater that $0$ on some region surrounding $P$.