# Graphical Interpretation of Continuity

## Trending Questions

**Q.**For sin−1[x], where [.] is greatest integer function, which of the following is/are true?

- Domain is [−1, 2)
- Product of all integral values of x in the domain is −1
- Sum of all integral values of x in the domain is 0
- Domain is {−1, 0, 1}

**Q.**Let sgn(y) and {y} denote signum function of y and fractional part function of y respectively.

Which of the following functions is (are) bijective?

- f:(−∞, 0]→(0, π2]defined by f(x)=sin−1(ex)
- f:[−1, 1]→{−1, 0, 1} defined by f(x)=sgn(sin−1|x|−cos−1|x|)
- f:[−3, 0]→[cos3, 1] defined by f(x)=cosx
- f:R−Z→R defined by f(x)=ln{x}

**Q.**If a, b and c are the integers that satisfy the domain of sin−1[x], where [.] is the greatest integer function, then

- a+b+c=0
- |a+b+c|=1
- abc=0
- |abc|=1

**Q.**

Let R be the feasible region (convex polygon) for a linear programming problem and let,

Z = + be the objective function. When Z has an optimal value (maximum or minimum), where the variables and are subject to constraints described by linear inequalities, this optimal value must occur at ____________of the feasible region.

corner point

either in the interior or on the boundary line

any point in the interior

any point on the bounadary line

**Q.**The relation R defined in A={1, 2, 3, } by a R b if ∣∣a2−b2∣∣≤5. Which of the following is false?

- R={(1, 1), (2, 2), (3, 3), (2, 1), (1, 2), (2, 3), (3, 2)}
- R−1=R
- Domain of R={1, 2, 3}
- Range of R={5}

**Q.**For any angle x∈[−π4, π3];cotx∈

- (−1, 1√3)
- (−∞, −1]∪[1√3, ∞)
- [−∞, 1√3)
- (−∞, −1)

**Q.**Given the graph of y=2tan(x+π4) ∀ y∈R.

Then, select the correct statement(s).

- For x∈[−2π, 2π], y=0 at:

x=−5π4, −π4, 3π4, 7π4 - Domain: R−(8n+1)π/4; n∈Z
- Domain: R−(4n+1)π/4; n∈Z
- For x∈[−2π, 2π], y=0 at:

x=−5π4, −π4, 3π4, 5π4

**Q.**

Among which of these points is f(x)=tanx discontinuous?

−π4

0

π4

π2

**Q.**

Among which of these points is f(x)=tanx discontinuous?

−π4

π4

π2

0

**Q.**Let f : R → R be a function. Define g:R → by g(x) = |f(x)| for all x. Then g is

- Onto if f is onto
- One-one if f is one-one
- Continuous if f is continuous
- None of these

**Q.**Let f : R → R be a function. Define g:R → by g(x) = |f(x)| for all x. Then g is

- Onto if f is onto
- One-one if f is one-one
- Continuous if f is continuous
- None of these

**Q.**In which of the following interval(s) of x,

cosx>0 ∀ x∈(π, 6π)

- (7π2, 9π2)
- (11π2, 6π)
- (3π2, 5π2)∪(7π2, 9π2)∪(11π2, 6π)
- (3π2, 5π2)

**Q.**

The function f(x)=tanx where x∈(−π4, π4).

is continuous

is discontinuous

is increasing

is decreasing

**Q.**

The function f(x)=tanx where x∈(−π4, π4).

is continuous

is discontinuous

is increasing

is decreasing

**Q.**Let f(x)=cot–1(sinx)+sin–1(x–{x}), where {x} denotes the fractional part function. Then

- f is one-one
- Df=[−1, 2)
- f is many one
- Rf={π4, π2, 3π4}

**Q.**Let f:N→N be defined by f(x)=x2+x+1, x∈N. Then f is

- many-one onto
- one-one but not onto
- one-one onto
- none of these

**Q.**Let f(x)=sin−1|sinx|+cos−1(cosx), x∈R. Which of the following statements is/are TRUE?

- f(f(3))=π
- f(x) is periodic with fundamental period 2π.
- f(x) is neither even nor odd.
- Range of f(x) is [0, 2π].

**Q.**If f(x)=[9x−3x+1], x∈(−∞, 1), then number of integers in the range of f(x) is (where [.] denotes greatest integer function)

**Q.**For the curve f(x)=11+x2, let two points on it be A(α, f(α)), B(−1α, f(−1α))(α>0). Find the minimum area bounded by the line segments OA, OB and f(x), where 'O' is the origin.

- (π−1)2
- π2
- (π−2)2
- Maximum area is always infinite

**Q.**Let f(x)=sinπxx2, x>0.

Let x1<x2<x3<...<xn<... be all the points of local maximum of f and y1<y2<y3<...<yn<... be all the points of local minimum of f.

Then which of the following options is/are correct?

- x1<y1
- xn+1−xn>2 for every n
- xn∈(2n, 2n+12) for every n
- |xn−yn|>1 for every n

**Q.**ntIntegrate the following with respect to x.n ntx/(x+7x+12)n

**Q.**

Find the number of discontinuities of the given function between x = 0 and x =2.

**Q.**The number of points of discontinuity and non-differentiability of the function f(x) in its domain are respectively

- 2, 1
- 2, 2
- 1, 1
- 1, 2

**Q.**For the interval [−2π, 2π], cosx>0 in the interval

- (−2π, −π) ∪ (π, 2π)
- (−2π, −3π2) ∪ (−π2, π2) ∪ (3π2, 2π)
- [−2π, −π2] ∪ [π2, 2π]
- [−π2, π2]

**Q.**Verify associativity for the following three mappings : f : N → Z

_{0}(the set of non-zero integers), g : Z

_{0}→ Q and h : Q → R given by f(x) = 2x, g(x) = 1/x and h(x) = e

^{x}.

**Q.**If f(x)=[sin πx] in (−12, 12), then which of the following(s) is/are correct:

where [x] denotes the greatest integer less than or equal to x.

- f(x) is continuous at x=0
- f(x) is continuous in (−1/2, 0)
- f(x) is continuous in (−1/2, 1/2)
- f(x) is continuous in (0, 1/2)

**Q.**Number of points of extremum of the function f(x)=|||x−1|−2|−3|, x ϵ [0, ∞) is

**Q.**

The function f(x)=tanx where x∈(−π4, π4).

is continuous

is discontinuous

is increasing

is decreasing

**Q.**Let f(x)=sinπxx2, x>0.

Let x1<x2<x3<...<xn<... be all the points of local maximum of f and y1<y2<y3<...<yn<... be all the points of local minimum of f.

Then which of the following options is/are correct?

- xn+1−xn>2 for every n
- |xn−yn|>1 for every n
- x1<y1
- xn∈(2n, 2n+12) for every n