I have 3 questions:

1. Explain the how the line, underlined by RED is formed, and how it leads to the line underlined by BLUE below.
2. Explain the meaning of the sentence underlined by GREEN.
3. Explain the difference between ?C1? class of functions and ?C?? class of functions

Please explain clearly showing each step as thoroughly as possible.

If you are using hand-written notes, then please ensure they are tidy and legible as untidy written notes are difficult to interpret.

Alternatively use LaTeX.

The following observations are immediate from the definitions. A function f: S C RR" - R" is differentiable at x E S if and only if each of the m component functions f': S -> R of f is differentiable at x, in which case we have Df (x) = (Df (x). ..., Df" (x)). Moreover, f is C on S if and only if each fi is C' on S. The difference between differentiability and continuous differentiability is non- trivial. The following example shows that a function may be differentiable every- where, but may still not be continuously differentiable. Example 1.51 Let f: R - R be given by 0 if x = 0 f ( x ) = x2 sin (1/x2) ifx 0.For x # 0, we have f'(x) = 2x sin () - (2)cos (+2). Since | sin(.) |