A). Let M be the set of functions defined on [0,1] that have a continuous derivative there ( one-sided derivatives at the endpoints).
Let p(x,y) = max_[0,1]|x'(t) - y'(t)|.

B). Let M be the set of continuous functions on [0,1] and define
p(x,y) = integral from 0 to 1 of |x(t) - y(t)|dt. Does this define a metric space? ( Also a proof here please for the yes or no answer).

Solution Preview

A.1
By definition, a metric d(x,y) must satisfy condition d(x, y) = 0 if and only if x = y.

It fails for p(x,y) = max_[0,1]|x'(t) - y'(t)|, because two functions can differ by a constant but still have p(x,y) = 0.

Therefore (M,p) is not a metric space

A.2
The definition of metric contains four conditions:
d(x, y) ≥ 0 (non-negativity) ...

... Any ∈ , can be approximated by a straight-segments function like that in ... space is defined on page 39 as the set of all functions continuous on interval ...

... A). Note: We assumed during this judgement that the function f(x) is not only continuous, but differentiable ... We divide now the interval (a, b) in a set of (n ...

... points of the following functions as x ranges over the sets of numbers ... f (n) (0) where f is each of the functions (1) sin ... (Q.1) It is said that a function f : R ...

... discontinuous point, then it is not well-bahaved compared with continuous function. ... find the intersection points of the two functions f and g . We set f ( x ...

... A graphing utility is used to graph the function f(x ... Zoom and trace functions of the utility are then used to ... The settings to do this are shown in window below. ...

... is used to see if a given set of data ... Un(a,b), if its probability density function is constant ... The cumulative distribution functions of two independent random ...

... Let f:Z->Z , where Z is the set of integers ... Prove that f(x)=x^2-4 is a continuous function. 9. Find derivatives of the following functions using differentiation ...

... probability distribution functions, fuzzy set, and order ... be specified to bound the corresponding monotonic functions. ... to constrain the values a function f can ...

... If you integrate this again then you end up with a continuous wave function. ... We must not set the derivative of psi_{1} at x = -a to zero, because the ...