Question

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.

Answer #1

Please right in comment box if doubt is still there.

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 .

Prove or provide a counterexample
Let f:R→R be a function. If f is T_U−T_C continuous, then f is
T_C−T_U continuous.
T_U is the usual topology and T_C is the open half-line
topology

if f: D - R be continuous, and D is close, then F(D) is closed.
prove or give counterexample

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 .

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

Theorem 4 states “If f is differentiable at a, then f is
continuous at a.” Is the converse also true? Specifically, is the
statement “If f is continuous at a, then f is differentiable at a”
also true? Defend your reasoning and/or provide an example or
counterexample (Hint: Can you find a graphical depiction in the
text that shows a continuous function at a point that is not
differentiable at that point?)

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...

NOTE- If it is true, you need to prove it and If it is
false, give a counterexample
f : [a, b] → R is continuous and in the open interval (a,b)
differentiable.
a) f rises strictly monotonously ⇐ ∀x ∈ (a, b) : f ′(x) > 0.
(TRUE or FALSE?)
b) f is constant ⇐⇒ ∀x∈(a,b): f′(x)=0 (TRUE or FALSE?)
c) If f is reversable, f has no critical point. (TRUE or
FALSE?)
d) If a is a “minimizer”...

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

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

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