Question

If a function $y=\varphi \left(x\right)$ is defined on [a, b] and $\varphi \left(a\right)\varphi \left(b\right)<0$ then

• A. such that if and only if is continuous
• B. a function differentiable on R-{0} satisfying the given hypothesis
• C. If satisfying the given hypothesis then must be discontinuous
• D.

Answer 1

The correct option is B a function differentiable on R-{0} satisfying the given hypothesis

Consider the function $\varphi \left(x\right)=\{\begin{array}{ccc}\frac{1}{x},& if& x\ne 0\\ 1,& if& x=0\end{array}definedon[-1,1],clearly\varphi (-1)\times \varphi \left(1\right)<0,and\varphi \left(x\right)isdifferentiableonR1\left\{0\right\}Butthereisnopointcϵ[-1,1]\ni \varphi \left(c\right)=0$