Continuity of Composite Functions

Q. Let $f:R\to R$ and $g:R\to R$ be defined as
$f\left(x\right)=\{\begin{array}{cc}x+a,& x<0\\ |x-1|,& x\ge 0\end{array}$ and $g\left(x\right)=\{\begin{array}{cc}x+1,& x<0\\ (x-1{)}^2+b,& x\ge 0\end{array},$
where $a,b$ are non-negative real numbers. If $\left(gof\right)\left(x\right)$ is continuous for all $x\in R,$ then $a+b$ is equal to

Q. If f(x) is continuous and g(x) is discontinuous at x = a, then the product function h(x) = f(x)g(x) must be discontinuous at x = a.

1. False
2. True

Q. If $f\left(x\right)=2x$ and $g\left(x\right)=\frac{x^2}{2}+1$ , then which of the following can be a discontinuous function

1. $f\left(x\right)+g\left(x\right)$
2. $f\left(x\right)–g\left(x\right)$
3. $f\left(x\right).g\left(x\right)$
4. $\frac{g\left(x\right)}{f\left(x\right)}$
5. $\frac{f\left(x\right)}{g\left(x\right)}$

Q. The composition of two continuous function is a continuous function

1. True
2. False

Q. If $f\left(x\right)=\frac{1}{(x-1)(x-2)}andg\left(x\right)=\frac{1}{x^2}$, then points of discontinuity of f(g(x)) are

1. 
2. 
3. 
4. 

Q. If $f\left(x\right)=\frac{1}{(x-1)(x-2)}andg\left(x\right)=\frac{1}{x^2}$, then points of discontinuity of f(g(x)) are

1. $\{-1,0,1,\frac{1}{\sqrt{2}}\}$
2. $\{-\frac{1}{\sqrt{2}},-1,0,1,\frac{1}{\sqrt{2}}\}$
3. $\{0,1\}$
4. $\{0,1,\frac{1}{\sqrt{2}}\}$

Q. If $f\left(x\right)=2x$ and $g\left(x\right)=\frac{x^2}{2}+1$ , then which of the following can be a discontinuous function

1. $f\left(x\right)+g\left(x\right)$
2. $f\left(x\right)–g\left(x\right)$
3. $f\left(x\right).g\left(x\right)$
4. $\frac{g\left(x\right)}{f\left(x\right)}$
5. $\frac{f\left(x\right)}{g\left(x\right)}$

Q. The composition of two continuous function is a continuous function

1. True
2. False

Q. If f(x) is continuous and g(x) is discontinuous at x = a, then the product function h(x) = f(x)g(x) must be discontinuous at x = a.

1. False
2. True

Q. The composition of two continuous function is a continuous function

1. True
2. False