Question

Prove that f(x)=x*cos(1/x) is continuous at x=0.

please give detailed proof. i guess we can use squeeze
theorem.

Answer #1

(i) Use the Intermediate Value Theorem to prove that there is a
number c such that 0 < c < 1 and cos (sqrt c) = e^c- 2.
(ii) Let f be any continuous function with domain [0; 1] such
that 0smaller than and equal to f(x) smaller than and equal to 1
for all x in the domain. Use the Intermediate Value Theorem to
explain why there must be a number c in [0; 1] such that f(c)
=c

Let a > 0 and f be continuous on [-a, a]. Suppose that f'(x)
exists and f'(x)<= 1 for all x2 ㅌ (-a, a). If f(a) = a and f(-a)
=-a. Show that f(0) = 0.
Hint: Consider the two cases f(0) < 0 and f(0) > 0. Use
mean value theorem to prove that these are impossible cases.

Prove that {n^3} does not converge to any number.
please give me detailed proof, thanks

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.

Is
the function continuous on (-infinity, infinity)? Explain.
f(x) = cos x; if x<0
0; if x= 0
1-x^2 ; if x>0

Use the Mean Value Theorem and the fact that for f(x) = cos(x),
f′(x) = −sin(x), to prove that, for x, y ∈ R,
| cos x − cos y| ≤ |x − y|.

Prove that if f(x) is a continuous function and f(x) is not zero
then g(x) = 1/f(x) is a continuous function.
Use the epsilon-delta definition of continuity and please
overexplain and check your work before answering.

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 .

Suppose that X is continuous random variable with PDF f(x) and
CDF F(x). (a) Prove that if f(x) > 0 only on a single (possibly
infinite) interval of the real numbers then F(x) is a strictly
increasing function of x over that interval. [Hint: Try proof by
contradiction]. (b) Under the conditions described in part (a),
find and identify the distribution of Y = F(x).

. Let f and g : [0, 1] → R be continuous, and assume f(x) = g(x)
for all x < 1. Does this imply that f(1) = g(1)? Provide a proof
or a counterexample.

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