Question

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

Answer #1

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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?)

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.

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 .

Let f(x) be a continuous, everywhere differentiable function.
What kind information does f'(x) provide regarding f(x)?
Let f(x) be a continuous, everywhere differentiable function.
What kind information does f''(x) provide regarding f(x)?
Let f(x) be a continuous, everywhere differentiable function.
What kind information does f''(x) provide regarding f'(x)?
Let h(x) be a continuous function such that h(a) = m and h'(a) =
0. Is there enough evidence to conclude the point (a, m) must be a
maximum or a minimum?...

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...

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

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 --> R be a differentiable function such that f' is
bounded. Show that f is uniformly continuous.

generate a continuous and differentiable function with the
following properties: f(x) is decreasing at x=-5, local minimum is
x=-3, local maximum is x=3

PROVE USING IVT. Suppose f is a differentiable function on [s,t]
and suppose f'(s) > 0 > f'(t). Then there's a point p in
(s,t) where f'(p)=0.

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