Question

If $f\left(x\right)$ and $g\left(x\right)$ are both continuous at $x=c$ then which of the following is/are always continuous at $x=c$?

• A. $f\left(x\right)+g\left(x\right)$
• B. $\left(f\right(x)-g(x\left)\right)\times f\left(x\right)$
• C. $g\left(x\right)\times f\left(x\right)$
• D. $\frac{f\left(x\right)-g\left(x\right)}{g\left(x\right)}$

Answer 1

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

The correct options are
A $f\left(x\right)+g\left(x\right)$
B $\left(f\right(x)-g(x\left)\right)\times f\left(x\right)$
C $g\left(x\right)\times f\left(x\right)$

$f\left(x\right)$ and $g\left(x\right)$ are continuous at $x=c$.

So $f+g,f-g,f\times g,$ are continuous

$(f-g)\times f$ is also continuous.

But $\frac{f-g}{g}$ may not be continuous because $g\left(c\right)$ should not be equal to zero and should be given in question.

without that condition, we cannot ascess anything.
For example: Let $f=2x^2$ and $g=x-3$ then both $f$ and $g$ are continuous at $3$ but $\frac{2x^2-(x-3)}{x-3}$ is not continuous at $x=3$ as denominator is $0$ in this situation.