Question

If $f$ and $g$ be two real functions continuous at all real number then,

• A. $f+g$ is continuous at $x=c$
• B. $fg$ is continuous at $x=c$
• C. $f\cdot g$ is continuous at $x=c$
• D. $\frac{f}{g}$ is continuous at $x=c$ , (provided $g\left(c\right)\ne 0$)

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $f\cdot g$ is continuous at $x=c$
B $fg$ is continuous at $x=c$
C $f+g$ is continuous at $x=c$
D $\frac{f}{g}$ is continuous at $x=c$ , (provided $g\left(c\right)\ne 0$)

Given: f and g are two real functions continue on all real numbers

To find: continuity of $(f+g),\left(fg\right),\frac{f}{g}andf\left(g\right(x\left)\right)atx=c$

Sol: ${lim}_{x\to c^{-}}f\left(x\right)={lim}_{x\to c^{-}}f\left(x\right)=f\left(c\right)$

Similarly ${lim}_{x\to c^{-}}g\left(x\right)={lim}_{x\to c^{+}}g\left(x\right)=g\left(c\right)$

We'll look for options now

$(f+g)$

${lim}_{x\to c^{-}}\{f\left(x\right)+g\left(x\right)\}={lim}_{x\to c^{-}}f\left(x\right)+{lim}_{x\to c^{-}}g\left(x\right)=f\left(c\right)+g\left(c\right)$

${lim}_{x\to c^{+}}\{f\left(x\right)+g\left(x\right)\}={lim}_{x\to c^{+}}f\left(x\right)+{lim}_{x\to c^{+}}g\left(x\right)=f\left(c\right)+g\left(c\right)$

$LHL=RHL⟹(f+g)$ is continuous

$f.g$

${lim}_{x\to c^{-}}f\left(x\right).g\left(x\right)=f\left(c\right).g\left(c\right)={lim}_{x\to c^{+}}f\left(x\right).g\left(x\right)$

$⟹f.g$ is continuous

${lim}_{x\to c^{-}}f\left(g\left(x\right)\right)=f\left(g\right(c\left)\right)={lim}_{x\to c^{+}}f\left(g\right(x\left)\right)$

$⟹fg$ is continuous

$\frac{f}{g}$

${lim}_{x\to c^{-}}\frac{f\left(x\right)}{g\left(x\right)}={lim}_{x\to c^{+}}\frac{f\left(x\right)}{g\left(x\right)}=\frac{f\left(c\right)}{g\left(c\right)}$

$⟹\frac{f}{g}$ is continous for all $CϵR$ except when $g\left(c\right)=0$

Hence, $(f+g),fg,f.g$ and $\frac{f}{g}$ all are continuous for all $C\epsilon R$