Question

Among various properties of continuous, we have if f is continuous function on $[a,b]$ and $f\left(a\right)f\left(b\right)<0,$ then there exists a point c in (a, b) such that $f\left(x\right)=0$ equivalently if f is continuous on $[a,b]$ and $x\in R$ is such that $f\left(a\right)<x<f\left(b\right)$ then there is $c\in (a,b)$ such that $x=f\left(c\right)$.

It follows from the above result that the image of a closed interval under a continuous function is a closed interval.

Let $f\left(x\right)=x,x\ne 0$ and $f\left(0\right)=1$. Then:

• A. f is a continuous function
• B. Range of f is an interval
• C. Range of f is R
• D. hypothesis of the result in the comprehension is violated

Answer 1

The correct option is D hypothesis of the result in the comprehension is violated

Let $f\left(x\right)=x,x\ne 0$ & $f\left(0\right)=1$

But as stated in question, if f is continuous on $[a,b],xϵR,\Rightarrow f\left(a\right)<x<f\left(b\right)$ there is $cϵ(a,b)$ such that $x=f\left(x\right)$.

So the assumption violates the continuity rule as if $x\ne 0,f\left(0\right)$ cannot be there equal to $1$ if it is continuous because $f\left(1\right)=1$. So the graph of assumed function breaks at $x=0$.