Question

Which of the following is/are not true?

• A. if $f\left(x\right)+g\left(x\right)$ is continuous at $x=a$, then $f\left(x\right)$ and $g\left(x\right)$ are both seperately continuous at $x=a$.
• B. If $f\left(x\right).g\left(x\right)$ is continuous at $x=a$, then $f\left(x\right)$ and $g\left(x\right)$ are seperately continuous at $x=a$.
• C. If $f\left(x\right).g\left(x\right)$ and $f\left(x\right)$ are discontinuous at $x=a$ then $g\left(x\right)$ is continuous at $x=a$.
• D. If both $f\left(x\right)$ and $g\left(x\right)$ are discontinuous at $x=a$ then $f\left(x\right)+g\left(x\right)$ may not be discontinuous at $x=a$.

Answer 1

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

The correct options are
A if $f\left(x\right)+g\left(x\right)$ is continuous at $x=a$, then $f\left(x\right)$ and $g\left(x\right)$ are both seperately continuous at $x=a$.
B If $f\left(x\right).g\left(x\right)$ is continuous at $x=a$, then $f\left(x\right)$ and $g\left(x\right)$ are seperately continuous at $x=a$.
C If $f\left(x\right).g\left(x\right)$ and $f\left(x\right)$ are discontinuous at $x=a$ then $g\left(x\right)$ is continuous at $x=a$.

Let $f\left(x\right)=\left[x\right],g\left(x\right)=x$, where $[.]\&.$ are the greatest integer & the fraction part function, respectively.
$\to f\left(x\right)+g\left(x\right)=x$, continuous at $x=0$ but $f\left(x\right)$ and $g\left(x\right)$ are not continuous at $x=0$.

$\to \left[x\right].\left\{x\right\}$ at $x=1$ is continuous but $\left[x\right]$ & $\left\{x\right\}$ are not seperately continuous.

$\to \left[x\right].\left\{x\right\}$ is discontinuous at $x=2$. and
$\left[x\right]$ is discontinuous at $x=2$
$\left\{x\right\}$ is discontinuous at $x=2$

$\to \left[x\right]$ and $\left\{x\right\}$ are discontinuous at $x=0$, but $\left[x\right]+\left\{x\right\}=x$ is continuous at $x=0$.