Question

A function is given by $f\left(x\right)=\log x.e^x$. Then $f\left(x\right)$ is

• A. Continuous throughout its domain
• B. Discontinuous at 2 points
• C. Discontinuous at 1 point
• D. Discontinuous at every real number

Answer 1

The correct option is A Continuous throughout its domain

Here the give function is,

$f\left(x\right)=\log x.e^x$

This can be understood by using the algebra of continuous functions.

If we have 2 functions $g\left(x\right),h\left(x\right)$ such that both are continuous on a given interval then $f\left(x\right)=g\left(x\right)\times h\left(x\right)$ will also be continuous on the same interval.

Here,

$f\left(x\right)=\log x.e^x$

We know that $\log x$ is continuous in its domain as well as the function $e^x$.

$f\left(x\right)$ is defined in the domain $(0,\infty )$ and through the above property we get that $\log x.e^x$ is also continuous in the same interval.