Question

If h(x) = g(x) + f(x) is a continuous function in a given interval then g(x) and f(x) individually will also be continuous in the same interval.

• A. True
• B. False

Answer 1

The correct option is B False

When we deal with algebra of continuous functions always keeps I mind that this relation is true only in reverse i.e., if f(x) and g(x) are continuous at a point or interval then $f+g,f-g,f.g,\frac{f}{g}(g\ne 0$) are also continuous. But the reverse is not true.

We can have examples which can show this.

$h\left(x\right)=x$;

$g\left(x\right)=\frac{1}{x}$

$f\left(x\right)=x-\frac{1}{x}$

here

$h\left(x\right)=g\left(x\right)+f\left(x\right)$ is continuous at every x $ϵ$ R but f(x) and g(x) individually taken are not continuous throughout as they are discontinuous at x=0.