Question

Part I) Prove that if f and g are continuous at a, then f+g is continuous at a using the epsilon-δ definition.

Part II) Let a, L ∈ R. Prove that if a ≥ L− ε for all positive, then a ≥ L.

Answer #1

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.

Use the ε-δ-definition of continuity to show that if f, g : D →
R are continuous then h(x) := f(x)g(x) is continuous in D.

Use ε − δ definition to prove that the function f (x) =
2x/3x^2 - 2 is continuous at the
point p = 1.

Let D ⊆ R, a ∈ D, let f, g : D −→ R be continuous functions. If
limx→a f(x) = f(a) and limx→a g(x) = g(a) with f(a) < g(a), then
there exists δ > 0 such that x ∈ D, 0 < |x − a| < δ =⇒
f(x) < g(x).

We know that any continuous function f : [a, b] → R is uniformly
continuous on the finite closed interval [a, b]. (i) What is the
definition of f being uniformly continuous on its domain? (This
definition is meaningful for functions f : J → R defined on any
interval J ⊂ R.) (ii) Given a differentiable function f : R → R,
prove that if the derivative f ′ is a bounded function on R, then f
is uniformly...

Show that if f and g are uniformly continuous on some interval I
then cf (for all c ∈ R) and f − g are all uniformly continuous on
I

Let f, g : [a, b] ---> R continuous such that
(f(a) - g(a)) (f(b) - g(b)) < 0.
a) Show that sup{|f(x) - g(x)| : x ϵ [a, b]} is strictly
positive and
achieved (is a maximum).

Apply ε − δ definition to show that f (x) = 1/x^2 is continuous
in (0, ∞).

Let f : [a,b] → R be a continuous function such that f(x)
doesn't equal 0 for every x ∈ [a,b].
1) Show that either f(x) > 0 for every x ∈ [a,b] or f(x) <
0 for every x ∈ [a,b].
2) Assume that f(x) > 0 for every x ∈ [a,b] and prove that
there exists ε > 0 such that f(x) ≥ ε for all x ∈ [a,b].

Using only definition 4.3.1 (continuity), prove that
f(x)=x2+3x+4 is continuous on R.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 3 minutes ago

asked 9 minutes ago

asked 22 minutes ago

asked 25 minutes ago

asked 27 minutes ago

asked 31 minutes ago

asked 37 minutes ago

asked 39 minutes ago

asked 51 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago