Pure Mathematics : Analysis

First of all answer the following question Quest-1 How we study calculus ?

Most of the time we use the results like definition, proposition and theorems to compute results. So, we can say that calculus deals with training students like us to able to master how do i compute ?

Example-1 : In calculus how do we compute derivative of any function ? like $\frac{d}{dx} (x^2) = 2x$. But, we don’t talk about the validation of the results we computed i.e why the procedure that we followed actually valid to compute the derivatives ? Therefore, In analysis we study why certain computations is valid ?

Quest-2 : How we study analysis ?

In analysis, we always ask questions like

1. What exactly does derivative mean?
2. Why does the limit exist?
3. For which functions does differentiation fail?
4. What assumptions are hidden? such thinking mindset we needed to study analysis ?

In mathematics we study mathematical objects , relationship between objects and how relationship between objects changes w.r.t to others and many more lead to the special class of mathematical structures that mathematicians are interested to explore through geometry, algebra and analysis etc.