# Relation between Continuity and Differentiability

## Trending Questions

**Q.**

What if the determinant is zero?

**Q.**Find the derivative of the function f(x)=2x2+3x−5 at x=−1. Also prove that f′(0)+3f′(−1)=0

**Q.**

Let f:[−12, 2]→R and g:[−12, 2]→R be functions defined by

f(x)=[x2−3] and g(x)=|x|f(x)+|4x−7|f(x), where [y] denotes the greatest integer less than or equal to y for yϵR. Then

- f is discontinuous exactly at three points in [−12, 2]
- f is discontinuous exactly at four points in [−12, 2]
- g is not differentiable exactly at four points in (−12, 2)
- g is not differentiable exactly at five points in (−12, 2)

**Q.**Let f={(1, 1), (2, 3), (0, −1), (−1, −3)} be a linear function from Z into Z. Find f(x)

**Q.**Let f(x)=tan(π[x−π])1+[x]2, where [.] denotes the greatest integer function. Then

- f(x) is continuous and differentiable at all x∈R
- f(x) is continuous at all x∈R but not differentiable at infinitely many points

- f(x) is neither continuous nor differentiable at a finite number of points

- f(x) is neither continuous nor differentiable at infinitely many points

**Q.**Let f:R→R be a function such that |f(x)|≤x2, for all x∈R. At x=0, f is

- Continuous but not differentiable

- Continuous as well as differentiable
- Neither continuous nor differentiable
- Differentiable but not continuous

**Q.**Let dydx−2ycotx=cosx such that y(π2)=0. If the maximum value of y is k, then the value of k is

**Q.**

If function be f(x)={x2−1x−1 when x≠1k when x=1 is continuous at x=1, then the value of k is?

- −1
- −2
- 2
- −3

**Q.**If f(x)={x2(sgn [x])+{x} , 0≤x<2sinx+|x−3| , 2≤x<4,

where [.] and {.} represent the greatest integer and the fractional part function, respectively. Then

- f(x) is differentiable at x=1
- f(x) is continuous but non-differentiable at x=1
- f(x) is non-differentiable at x=2
- f(x) is discontinuous at x=2

**Q.**The number of point(s) of non-differentiability for f(x)=[ex]+|x2−3x+2| in (−1, 3) is ( where [.] denotes greatest integer function, e3=20.1 )

**Q.**If sin−1x+|y|=2y, then y as a function of x is

- defined for −1≤x≤1
- continuous at x=0
- differentiable for all x in (−1, 1)
- such that dydx=13√1−x2 for −1<x<0

**Q.**Let S={(λ, μ)∈R×R:f(t)=(|λ|e|t|−μ)⋅sin(2|t|), t∈R, is a differentiable function}.

Then S is a subset of :

- R×[0, ∞)
- [0, ∞)×R
- R×(−∞, 0)
- (−∞, 0)×R

**Q.**Given f(x) is continuos at x

_{0}, for f(x) to be differentiable at x

_{0}, the left hard Derivative and the right hand Derivative must exist fanitely.

- False
- True

**Q.**

Examine if Rolleâ€™s Theorem is applicable to any of the following functions. Can you say some thing about the converse of Rolleâ€™s Theorem from these examples?

(i)

(ii)

(iii)

**Q.**Let f(x)={1|x|:|x|≥1ax2+b:|x|<1 be continuous and differentiable every where. Then a and b are:

- −12, 32
- 12, −32
- 12, 32
- 13, 12

**Q.**At the point x = 1, the given function f(x)={x3−1;1<x<∞x−1;−∞<x≤1 is

- Discontinuous and not differentiable
- Continuous and differentiable
- Continuous and not differentiable
- Discontinuous and differentiable

**Q.**If f(x)=(x−1)2, then which of the following is/are true

- f is continuous at x=1
- f is not derivable at x=1
- f is discontinuous at x=1
- f is derivable at x=1

**Q.**

Find the value of limx→π4cos x−sin xcos 2 x

limit does not exist

**Q.**If f(x)=⎧⎨⎩1, when0<x≤3π42sin29x, when3π4<x<π , then

- f(x) is continuous at x=3π4
- f(x) is discontinuous at x=3π4
- f(x) is continuous at x = 0
- f(x) is continuous at x = π

**Q.**Let f(x)=x|x|, g(x)=sin(x) and h(x)=(g∘f)(x). Then

- h(x) is not differentiable at x=0.

- h′(x) is continuous but not differentiable at x=0.

- h(x) is differentiable at x=0, but h′(x) is not continuous at x=0.

- h′(x) is differentiable at x=0.

**Q.**If f(x)=⎧⎪ ⎪⎨⎪ ⎪⎩x−1x<014, x=0x2, x>0 , then

- limx→0+f(x)=1
- f(x) is discontinuous at x = 0
- limx→0−f(x)=1
- None of these

**Q.**Consider f(x)=sgn(sinx)+{x};2≤x≤4 and g(x)=−2+∣x−3∣; where {.} denotes fractional part function and sgn denotes signum function. Then limx→3gof(x) is equal to

- -1
- 0
- 1
- does not exist

**Q.**The function defined by f(x)={|x−3|;x≥114x2−32x+134;x<1 is

- Continuous at x = 3
- Continuous at x = 1
- Differentiable at x = 1
All the above

**Q.**Paragraph for below question

नीचे दिए गए प्रश्न के लिए अनुच्छेद

Consider the function

*f*(

*x*) = |sin

*x*| + |cos

*x*|,

*x*∈

*R*.

फलन

*f*(

*x*) = |sin

*x*| + |cos

*x*|,

*x*∈

*R*पर विचार कीजिए।

Q. Value of f′(π4) is

प्रश्न - f′(π4) का मान है

- −√2
- √2
- 0
- 1√2

**Q.**If f(x)={x2−4x+3x2−1, for x≠12, for x=1 , then

- f(x) is discontinuous at x = 1
- limx→1+f(x)=2
- limx→1−f(x)=3
- None of these

**Q.**Examine the continuity of the function

$f\left(x\right)=\left\{\begin{array}{ll}3x-2,& x\le 0\\ x+1,& x>0\end{array}\mathrm{at}x=0\right.$

Also sketch the graph of this function.

**Q.**The function f(x)={14x−1, x≠00, x=0 is continuous

- everywhere except at x=0 and x=∞
- nowhere
- everywhere
- everywhere expect at x=0

**Q.**Given that f(x) = ⎧⎨⎩x.g(x)|x|, x≠00, x=0 g(0) = 0 = g'(0), then f'(0) equals

- 1
- -1
- 2
- 0

**Q.**If f(x)={x−|x|x, when x≠02, when x=0 , then

- limx→0f(x)=2
- None of these
- f(x) is continuous at x = 0
- f(x) is discontinuous at x = 0

**Q.**Prove that the Greatest Integer Function f:R→R given by f(x)=[x], is neither one-one nor onto, where [x] denotes the greatest integer less that or equal to x