I.3 Continuity and Differentiation; Pages 31 – 33.

One basic definition of a limit of a sequence of numbers an is that : whatever positive delta you choose, there is a number N such that for all bigger numbers n the difference between an and l (ell) is less than d.   This may be written mathematically as:  \forall d, N,  \forall n s N, |an – l| < d.   The idea of limit applies just as well when one has a distance between objects defined rather than the difference between numbers.  In this case there is a limiting object.  

One says that “the limit of the sequence an is l (ell)” or “”an converges to l”.  One can also say that “this happens as n tends to infinity”.  Any sequence that has a limit is called convergent.

We say that a function f is continuous at a if  \forall  > 0,  $ d > 0, (| x – a| < d \Rightarrow  |f(x) – f(a)| < e).  As with limits this idea can be extended to more general cases where distance between objects in X and a possibly different distance between objects in Y is known.  Then we have f is continuous at a if  \forall  > 0,  $ d > 0, ( distance(x,a) < d \Rightarrow  distance(f(x),f(a)) < e).  We also say f is continuous if f is continuous at every a.  It turns out that continuous functions are functions that conserve the structure of convergent sequences.  For a continuous function f, if an converges to x, an z x, then f(an) converges to f(x), f(an) z f(x).  

The derivative of a function f at a value a is the rate of change of f(x) as x passes through a.  One may think of this as the linear approximation to f at a.  This derivative idea of functions may be generalized to linear maps.  If u is a function from Rn to Rm then then u is said to be differentiable at x Î Rn if there is a linear map T: Rn to Rm such that u(x + h) = u(x)  + T(h) + e(h), with e(h) small relative to h.  The linear map T is the derivative of u at h.

A special case is with m=1.  The matrix of T is a row vector of length n which may be denoted as  \nabla \!\,f(x) and is referred to as the gradient of f at x.  This vector points in the direction in which f increases most rapidly and its magnitude is the rate of change in that direction.  

Leave a Reply

You must be logged in to post a comment.