The total number of functions $f: \{1, 2, 3, 4\} \to \{1, 2, 3, 4, 5, 6\}$ such that $f(1) + f(2) = f(3)$ is equal to:

Let $c, k \in R$. If $f(x)=(c+1) x^{2}+(1-c^{2}) x+2 k$ and $f(x+y)=f(x)+f(y)-x y$,for all $x, y \in R$,then the value of $|2(f(1)+f(2)+f(3)+\ldots+f(20))|$ is equal to

If $f(x) = \frac{2^x}{2^x + \sqrt{2}}$,$x \in R$,then $\sum_{k=1}^{81} f\left(\frac{k}{82}\right)$ is equal to :

Let $N$ be the set of positive integers. For all $n \in N$,let $f_n = (n+1)^{1/3} - n^{1/3}$ and $A = \{n \in N : f_{n+1} < \frac{1}{3(n+1)^{2/3}} < f_n\}$. Then,

Which of the following real-valued functions is/are not even functions?

Let $f, g: R \rightarrow R$ be functions defined by $f(x) = \begin{cases} [x] & x < 0 \\ |1-x| & x \geq 0 \end{cases}$ and $g(x) = \begin{cases} e^x - x & x < 0 \\ (x-1)^2 - 1 & x \geq 0 \end{cases}$ where $[x]$ denotes the greatest integer less than or equal to $x$. Then,the function $(f \circ g)(x)$ is discontinuous at exactly