Question

Prove or give a counter example: If f is continuous on R and differentiable on R ∖ { 0 } with lim x → 0 f ′ ( x ) = L , then f is differentiable on R .

Answer #1

Prove or give a counterexample: If f is continuous on R and
differentiable on R∖{0} with limx→0 f′(x) = L, then f is
differentiable on R.

give an example of a continuous and differentiable function at a
point x = a, with a null derivative at this point, f '(a) = 0, and
that f has neither a maximum nor a minimum at x = a

let F : R to R be a continuous function
a) prove that the set {x in R:, f(x)>4} is open
b) prove the set {f(x), 1<x<=5} is connected
c) give an example of a function F that {x in r, f(x)>4} is
disconnected

if
the function f is differentiable at a, prove the function f is also
continuous at a.

Let f : R → R be a bounded differentiable function. Prove that
for all ε > 0 there exists c ∈ R such that |f′(c)| < ε.

4a). Let g be continuous at x = 0. Show that f(x) = xg(x) is
differentiable at x = 0 and f'(0) = g(0).
4b). Let f : (a,b) to R and p in (a,b). You may assume that f is
differentiable on (a,b) and f ' is continuous at p. Show that f'(p)
> 0 then there is delta > 0, such that f is strictly
increasing on D(p,delta). Conclude that on D(p,delta) the function
f has a differentiable...

Let a < b, a, b, ∈ R, and let f : [a, b] → R be continuous
such that f is twice differentiable on (a, b), meaning f is
differentiable on (a, b), and f' is also differentiable on (a, b).
Suppose further that there exists c ∈ (a, b) such that f(a) >
f(c) and f(c) < f(b).
prove that there exists x ∈ (a, b) such that f'(x)=0.
then prove there exists z ∈ (a, b) such...

Let f be continuous on [ 0 , ∞ ) and differentiable on ( 0 , ∞ )
. If f ( 0 ) = 0 and | f ′ ( x ) | ≤ | f ( x ) | for all x > 0 ,
then f ( x ) = 0 for all x ≥ 0 .

Prove or give a counter-example:
(a) if R ⊂ S and T ⊂ U then T\ S ⊂ U \R.
(b) if R∪S⊂T∪U, R∩S= Ø and T⊂ R, then S ⊂ U.
(c) if R ∩ S⊂T ∩ S then R⊂T.
(d) R\ (S\T)=(R\S) \ T

Prove or give a counter example for "If E1 and
E2 are independent, then they are conditionally
independent given F."

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