I A Maron Solutions for Chapter: Introduction to Mathematical Analysis, Exercise 15: Additional Problems

Investigate the functions $f\left[g\left(x\right)\right]$ and $g\left[f\left(x\right)\right]$ for continuity if$f\left(x\right)=\mathrm{sgn}x$ and $g\left(x\right)=x\left(1-x^2\right)$

Prove that the function $f\left(x\right)=\left\{\begin{array}{cc}2x& \mathrm{at} -1\le x\le 0\\ x+\frac{1}{2}& \mathrm{at} 0<x\le 1\end{array}\right.$is discontinuous at the point $x=0$ and nonetheless has both maximum and minimum values on $\left[-1,1\right]$.

Given the function $f\left(x\right)=\left\{\begin{array}{l}\left(x+1\right)2^{-\left(\frac{1}{\left|x\right|}+\frac{1}{x}\right)},x\ne 0\\ 0,x=0\end{array}\right.$Ascertain that on the interval $\left[-2,2\right]$ the function takes on all intermediate values from $f\left(-2\right)$ to $f\left(2\right)$ although it is discontinuous (at what point?).

Prove that if the function $f\left(x\right):\left(1\right)$ is defined and monotonic on the interval $\left[a, b\right];\left(2\right)$ traverses all intermediate values between $f\left(a\right)$ and $f(b)$, then it is continuous on the interval $\left[a, b\right]$.

Let the function $y=f\left(x\right)$ be continuous on the interval $\left[a, b\right]$, its range being the same interval $a\le y\le b$. Prove that on this closed interval there exists at least one point $x$ such that $f\left(x\right)=x$. Explain this geometrically.

Prove that the equation $x{2}^x=1$ has at least one positive root which is less than unity.

Prove that if a polynomial of an even degree attains at least one value the sign of which is opposite to that of the coefficient at the superior power of $x$ of the polynomial, then the latter has at least two real roots.

Prove that the inverse of the discontinuous function $y=\left(1+x^2\right)$ $\mathrm{sign}\left(x\right)$ is a continuous function.