I am a recently graduated student and doing Post Graduation now. I often come across uniform convergence, uniform continuity etc. As we all know that we check continuity and convergence easily by just seeing the graph of function and it is very helpful in solving problems. But I just can't analyse uniform convergence and uniform continuity geometrically. An explanation would be really helpful.
If there is some book from which we can analyse concepts of analysis geometrically, I'd be very interested in reading it; please let me know if you know of such a text.
Thanks in advance.
In the standard definitions of function convergence and continuity, the size of the "error" (i.e., $\delta$ in continuity and the start of the tail, $N$, in convergence) depends on $x$. The hypothesis of uniformity removes the dependence on $x$, so that, given $\epsilon$, the same $\delta$ or $N$ work anywhere in the function's domain.
Geometrically, this means that your geometric picture of continuity works everywhere on the graph, without depending on where you're applying it, as in the video that hcl14 linked in the comments above. For convergence, there is a "band" about the function you're converging to such that the graphs of all the functions in the tail lie inside the band. This has some pictures of what I mean.