Question

If $f\left(x\right)$ is continuous at $x=c$ and $g\left(x\right)$ continuous at $x=c$, then which of the following is/are always continuous at $x=c$?
Note: $k$ is some scalar parameter.

• A. $k\ast \left(f\right(x)-g(x\left)\right)\ast f\left(x\right)$
• B. $\left(f\right(x)\ast g(x){)}^k$
• C. $\frac{f(x{)}^k}{g\left(x\right)}$
• D. None

Answer 1

Answer: (A), (B)

$k\ast \left(f\right(x)-g(x\left)\right)\ast f\left(x\right)$
$\left(f\right(x)\ast g(x){)}^k$

Given f and g are continuous at $x=c$

$\Rightarrow f+g,f-g,f\ast g,kf,kg,\frac{f}{g}\left(g\right(a)\ne 0)$ are continuous

A. $(kf\left(x\right)-kg\left(x\right))\ast f\left(x\right)$

$=k{\left(f\left(x\right)\right)}^2-kg\left(x\right)f\left(x\right)$

Let $h\left(x\right)=f\left(x\right)\ast f\left(x\right)$

$h\left(x\right)$ is continuous according to Algebra of continuous functions

Since h is continuous $kh\left(x\right)$ is also continuous (ACF)

Similarly $m\left(x\right)=g\left(x\right)\ast f\left(x\right)$ is also continuous

$\Rightarrow km\left(x\right)$ also continuous

$kh\left(x\right)-km\left(x\right)=k(h\left(x\right)-m\left(x\right))$

k(continuous function)$=$continuous function

B: Let $p\left(x\right)=f\left(x\right)\ast g\left(x\right)$

$p\left(x\right)$ is continuous (ACF)

$\Rightarrow p\left(x\right)\ast p\left(x\right)$ is continuous

$\Rightarrow p\left(x\right)\ast p\left(x\right)\ast ...k$ times is continuous

$\Rightarrow {\left(p\left(x\right)\right)}^k$ is continuous

C.$g\left(x\right)$ is given as continuous but $g\left(x\right)$ could be zero

$\Rightarrow \frac{{f\left(x\right)}^k}{g\left(x\right)}$ is not continuous.