# Tagged Questions

Elementary questions about functions, notation, properties, and operations such as function composition.

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### Applications for holomorphic functions?

Could anyone give me an insight into practical applications of holomorphic functions (I am using the term in the way in which it is related to Riemann's work)?
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### Uniform continuity of two functions

Investigate uniform continuity of the following functions: $$a) \ f(x)=\frac{1}{x} \\ b) \ f(x)=\cos \frac{1}{x}$$ How to deal with such questions, i have little knowledge about that topic thus i ...
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### $f(x)$ be differentiable and have a local minimum in $x_0$, show with definition $f'_+(x_0)\ge0, f'_-(x_0)\le0, f'(x_0)=0$

Let $f(x)$ be differentiable in $x_0$, $x_0$ is a local minimum, prove with the definition that $f'_+(x_0)\ge0, f'_-(x_0)\le0, f'(x_0)=0$. I get that $f$ is decreasing from the left and increasing ...
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### Small question: Name for the x of function f such that f(x)=x?

Background When doing maths and chemistry problems, I often came across things like $$x-\frac{x}{2}=\frac{x}{2}$$ It might seems trivial, but I found that it is often the presence of expressions like ...
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I was trying to understand the following: which I got from: http://www.mit.edu/~9.520/fall14/slides/class14/class14_manifold.pdf I was wondering, why the domain was: $$\alpha(U_{\alpha} \cap U_{... 0answers 38 views ### Interpretation of an integral of a function f When we think of a Riemann integral, it is usually defined as \lim_{\Delta x_{k}\rightarrow 0}\sum_{k = 1}^{n}f(x_{k}^{*})\Delta x_{k} = \int_{a}^{b}f(x)~dx. This means that f(x) should be ... 1answer 77 views ### Defining the differentiation operator The differentiation operator is the function \frac{\mathrm{d} }{\mathrm{d} x}: f \mapsto f'. My question is, does the operator really take an entire function f as an argument? For example, when ... 1answer 48 views ### Function as onto I have a doubt that if a function is one-to-one then it will also be onto. If a function f(x) is defined such that f: \mathbb{R} \rightarrow \mathbb{R} then if the function is many to one then ... 1answer 42 views ### Showing that f,g are invertible if A is a finite set and f,g: A\to A such that f\circ g is invertible Let A be a finite set and f,g: A\to A such that f\circ g is invertible. Prove f,g are invertible. Prove that if A is an infinite set, it doesn't mean that f,g are invertible. I ... 3answers 118 views ### Handling division by zero axiomatically Suppose we define the multiplicative inverse function on real numbers as follows: \forall{x \in \mathbb{R}}(x \neq 0 \implies x \times \frac{1}{x} = 1) . Consider this truth table. \begin{array} {... 0answers 31 views ### Finding out if a function is invertible: f,g:\mathbb N\to \mathbb N, g(x)=2x and f with cases Let f,g:\mathbb N\to \mathbb N such that g(x)=2x and f(x)=\begin{cases}\frac x 2 &, x\in\mathbb N_{even}\\ x+9 &,x\in\mathbb N_{odd}\end{cases} ... 1answer 81 views ### Recursively enumerable sets are domain of partial recursive functions My definition of recursively enumerable set is that it is the language recognized by some Turing machine. I want to show that this definition is equivalent to "a r.e. set is the domain of some ... 1answer 51 views ### How do I prove that the only possible function is exp? Let´s say we have a differentiable function f : \mathbb{R} -> \mathbb{R} with f' = f and f(0) = 1 . How do I show that the only possible function for this to work f = exp ? ... 2answers 309 views ### Number of one -to-one functions Let A = \{1, 2, 3, 4\} and B = \{a, b, c, d, e\}. what is the number of functions from A to B are either one-to-one or map the element 1 to c? My answer is 166, but I'm not really sure ... 1answer 131 views ### Can every function be a composite to itself and how to know if a composite between two functions is defined? Can every function be a composite to itself? like we have f:A\to B is f \circ f always defined? Can we say that if f is a injection/surjection/bijection then so is f\circ f? Also, how do ... 4answers 48 views ### Double branch \sqrt x or square function turned 90°? I have this idea for a graph but don't know what function could describe it better. The idea is something like the "squared" function turned 90 degrees to the right, so that possible values for x ... 2answers 157 views ### Math Contest Question with Polynomials Prove that there does not exist a polynomial f(x) with integer coeﬃcients for which f(2008) = 0 and f(2010) = 1867. This is a question from CMOQR (Qualifier for Canadian Math Olympiad , not the ... 0answers 26 views ### Is there a name/notation for coordinate-wise identical function? Let's define g: \mathbb{U}^n \rightarrow \mathbb{V}^n where \mathbb{U} and \mathbb{V} are arbitrary sets as$$g(u) = \left[f(u_1), f(u_2), \ldots, f(u_n) \right]^T$$for some f: \mathbb{U} \... 1answer 56 views ### Find the function continuous or discontinuous \sum_{n=1}^∞  (x+2)^n \over n! + x^2 , Interval = [1,2] Is this function continuous in that interval ? I tried but the factorials are troubles. 4answers 65 views ### Writing a piecewise function for f(x) = \mid x+3\mid -\mid x-1\mid  I am wanting to write a piecewise function for the following:$$f(x)= \mid x+3\mid -\mid x-1\mid $$I know how to write piecewise functions for functions that have a single set of absolute ... 1answer 94 views ### Graphing a Piecewise Function I graphed this function below. I want to make sure I am graphing piecewise functions such as this one correctly. 1answer 89 views ### Inverse function for y=\lfloor x\rfloor+x Find the inverse function of the following function: y=\lfloor x\rfloor +x I have tried writing down x as \lfloor x\rfloor +\{x\} but didn't get anywhere with that. A proper hint would ... 1answer 351 views ### Exponential function given two points I am trying to find an exponential function satisfying two points (having base "exp"). After some search, I couldn't find something relative (the most relevant was that https://www.youtube.com/watch?v=... 2answers 70 views ### Is f(x) =|x| - 3 even, odd, or neither?$$f(x)=|x|-3$$Is the function above odd, even or neither? I know that a function is even if f(x) = f(-x):$$f(-x) = |(-x)| - 3f(-x) = x-3$$Does this mean that the function is even? ... 1answer 20 views ### How to resemble hyperbolic trigonometric functions (HTF) from normal trigonometric functions(NTF)?? There are many properties of HTF similliar but little different than NTF, Is there some pattern or rule that makes me, who only knows NTF, able to get HTF from direct resemblence to NTF? 5answers 1k views ### Why does notation for functions seem to be abused and ambiguous? I really need to clear up a few things about function notation; I can't seem to grasp how to interpret it. As of right now, I know that a function is roughly a mapping between a set X and a set Y, ... 0answers 45 views ### Name for a nowhere constant function? Is there a pithy name for a function f : \mathbb{R}^n \rightarrow \mathbb{R}^m such that there is no non-degenerate interval I \subseteq \mathbb{R}^n such that f is constant on I (by 'f is ... 3answers 54 views ### Multiple variables calculus: condition for f to be continuous using curves Prove f:\mathbb{R}^n\to\mathbb{R} is continuous iff for every curve, \gamma:[a,b]\to\mathbb{R}^n: f\circ \gamma :\mathbb{R}\to\mathbb{R} is continuous. (\Rightarrow) is trivial. (\Leftarrow)... 4answers 95 views ### Proving a set is uncountable. [duplicate] I need to prove that the set of all functions \mathcal{F}:\mathbb{N}\rightarrow \left \{ 0,1 \right \} is uncountable. I'm not too sure at all how to do this. My initial idea was to try and show ... 2answers 3k views ### Is the function f(x) = tan(x) odd, even, or neither? Is the function f(x) = \tan(x) odd, even, or neither? Here is what I have so far: I know the function is not even because f(x) ≠ f(-x):$$f(-x) = \tan(-x)\tan(-x) ≠ \tan(x)$$Now I want ... 1answer 46 views ### Prove that f(x,y) is not continuous for any a element of R Given function$$ f(x,y) = \begin{cases} 3xy/(x^2 + y^2) & (x,y) \neq (0,0) \\ a & (x,y) = (0,0) \end{cases}$$prove there exists no a \in \Bbb R ... 1answer 51 views ### How to prove (2^{-1/y}(1-x)+x)^{-y} is increasing in y, when x,y \in (0,1). As the title suggests, how to prove (2^{-1/y}(1-x)+x)^{-y} is increasing in y when x,y \in (0,1)? 4answers 61 views ### limit of the function x \to 1 Why$$\lim_{x\rightarrow1} \frac{ e^{ \frac{1}{\pi} \ln x } -1} {\frac{1}{\pi} \ln x }=1 I do not get it at all... I do not want explanation that would involve use of l'hospital rule
Find two numbers $a$ and $b$ with $a \leq b$ such that $\int_a^b (6-x-x^2)dx$ has the largest value.
I'm stuck with the following question: Let P be a particle at point $(1,2)$ on the surface $z=x^2y^2$. At $t=0$ the particle is left and moves freely. Find the path that the particle passes during ...