Relation between Continuity and Differentiability

Q. The number of points where $f\left(x\right)=\{\begin{array}{cc}{(x-1)}^2\cos \left(\frac{1}{x-1}\right)-|x|,& x\ne 1\\ -1,& x=1\end{array}$ is not differentiable is

Q.

$f\left(x\right)=2\left|\frac{x-1}{x-1}\right|\forall x\&x\ne 1$, What should be the value of f(x) at x=1 so that it is differentiable from $(-\infty ,\infty )$

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Q.

If $f\left(x\right)=|lo{g}_{10}^x|\forall xϵ(0,\infty )$ then

1. f(x) is continuous in

2. f(x) is differentiable in (0, ∞)

3. f(x) is continuous but not differentiable in (0, ∞)

4. None of these

Q.

Is the function, f(x) = min {x, x²} ᵿx ϵ R continuous?

1. undefined
2. undefined
3. undefined
4. undefined

Q.

If h(x) = min {x, x²} for x ϵ R. Find LHD and RHD at x = 1.

1. undefined
2. undefined
3. undefined
4. undefined

Q. Given f(x) is continuos at x₀, for f(x) to be differentiable at x₀, the left hard Derivative and the right hand Derivative must exist fanitely.

1. False
2. True

Q.

If the function f(x) has LHD and RHD at x₀, which are finite and equal, then f(x) is differentiable at x₀, always.

1. True
2. False

Q.

Let $f:[-\frac{1}{2},2]\to R$ and $g:[-\frac{1}{2},2]\to R$ be functions defined by
$f\left(x\right)=[x^2-3]\text{and}g\left(x\right)=|x|f\left(x\right)+|4x-7|f\left(x\right)$, where [y] denotes the greatest integer less than or equal to y for $yϵR$. Then

1. f is discontinuous exactly at three points in $[-\frac{1}{2},2]$
2. f is discontinuous exactly at four points in $[-\frac{1}{2},2]$
3. g is not differentiable exactly at four points in $(-\frac{1}{2},2)$
4. g is not differentiable exactly at five points in $(-\frac{1}{2},2)$

Q. Let $f\left(x\right)=\{\begin{array}{cc}x{e}^{-(\frac{1}{|x|}+\frac{1}{x})},& x\ne 0\\ 0,& x=0\end{array}$ Then which of the followings is/are correct.

1. $f\left(x\right)$ is continuous at $x=0$
2. $f\left(x\right)$ is differentiable at $x=0$
3. $f\left(x\right)$ is not continuous at $x=0$
4. $f\left(x\right)$ is not differentiable at $x=0$

Q.

The function

$f\left(x\right)=\{\begin{array}{cc}|x-3|,& x\ge 1\\ \frac{x^2}{4}-\frac{3x}{2}+\frac{13}{4},& x<1\end{array}$,

is

1. continuous at $x=1$

2. differentiable at $x=1$

3. discontinuous at $x=1$

4. differentiable at $x=3$