Homework Statement
Prove that if f''(x) exists and is continuous in some neighborhood of a, than we can write
f''(a)= \lim_{\substack{h\rightarrow 0}}\frac{f(a+h)- 2f(a)+f(a-h)}{h^2}
The Attempt at a Solution
I just proved in the first part of the question, not posted, that...
Solved: Real Analysis, differentiation
Homework Statement
If g is differentiable and g(x+y)=g(x)(g(y) find g(0) and show g'(x)=g'(0)g(x)
The Attempt at a Solution
I solved g(0)=1
and
I got as far as
g'(x)=\lim_{\substack{x\rightarrow 0}}g(x) \frac{g(h)-1}{h}
but now I...
Homework Statement
Using a delta epsilon method prove:
\mathop {\lim }\limits_{x \to 1 } x^3+2x^2-3x+4= 4
The Attempt at a Solution
I got so far as breaking the equation into
=|x||x+3||x-1| now how do I bound it? Also, even more basic question, once I found the bound how do I put the...
Because there are infinitely many irrational numbers which would make the graph continuous on the irrationals, but on an interval there would be rationals mixed in between the irrationals?
Homework Statement
For each a\in\mathbb{R}, find a function f that is continuous at x=a but discontinuous at all other points.
The Attempt at a Solution
I guess I am not getting the question. I need to come up with a function, I was thinking of a piecewise defined one, half rational...
I know that the function is continuous at x=0. So how does showing it is continuous at zero help with showing the function with the property f(x_{1}+x_{2})=f(x_{1})+f(x_{2}) is continuous?
Thank you
Homework Statement
Prove: If f is defined on \mathbb{R} and continuous at x=0, and if f(x_{1}+x_{2})=f(x_{1})+f(x_{2}) \forall x_{1},x_{2} \in\mathbb{R}, then f is continuous at all x\in\mathbb{R}.
Homework Equations
None
The Attempt at a Solution
Need a pointer to get started...
Homework Statement
Describe the locus of points z satisfying the given equation.
Homework Equations
Im(2iz)=7
|z-i|=Re(z)
The Attempt at a Solution
I started on the second one:
I think that Re(z) is just x, then
I squared both sides, simplified and got
(y-1)^2=0 is this...
You mean that if \int \limits_a^b f(x) dx=0 and \int \limits_a^b g(x) dx=0, can I prove that \int \limits_a^b f(x)+g(x) dx=0? Also, if \int \limits_a^b f(x) dx=0 then k\int \limits_a^b f(x) dx=0?
Yet another problem I need to get some starting help on:
Show that the set of continuous functions f=f(x) on [a,b] such that \int \limits_a^b f(x) dx=0 is a subspace of C[a,b]
Thank you