The chapter on Continuity and Differentiability is essential because it forms the foundation for applying limits, derivative definitions, and powerful theorems like the Mean Value Theorem (MVT) and Rolle’s theorem. These ideas frequently appear in board exams for continuity/differentiability at points, and in competitive exams for more subtle functions involving oscillations, absolute values, and parameter-based conditions—where understanding exact definitions and theorem requirements is crucial.
25
Minutes
20
Questions
1 / -0
Marking
Q1. Let be continuous on and differentiable on with and . Which of the following is guaranteed by the Mean Value Theorem?
There exists such that
There exists such that
There exists such that
There exists such that
Q2. Let be defined by
where . For which values of and is differentiable at ?
and
with
for every real
for every real
Q3. Let for and , where . For which values of is differentiable at ?
Q4. Suppose is differentiable on and for every rational . Which of the following statements must hold?
is constant on
only at rational points
is constant only on the set of rational numbers
can have a jump discontinuity at some irrational point
Q5. Let
Which of the following is correct about and ?
is differentiable everywhere and is continuous at
is differentiable everywhere but is not continuous at
is not differentiable at
is unbounded in every neighbourhood of
Q6. Let for and . Which of the following is true about at ?
is continuous at but not differentiable at
is differentiable at and
is differentiable at and
is discontinuous at
Q7. Define for and . For which value of is continuous at , and what is ?
continuous iff , and
continuous iff , and
continuous iff , and
continuous iff , and does not exist
Q8. Let for and . Which statement is correct?
is differentiable for all and is not continuous at
is not differentiable at
is differentiable for all and is continuous at
is continuous at but not differentiable at infinitely many points arbitrarily close to
Q9. Define for and . Which of the following describes the behaviour of and near ?
is not differentiable at
exists everywhere and is bounded in some neighbourhood of
exists everywhere and is continuous at
, exists for all , but is unbounded in every neighbourhood of
Q10. Let with real . Which statement about differentiability at is correct?
is differentiable at iff , and exists iff
is differentiable at iff , and exists iff
is differentiable at for every , and exists iff
is differentiable at iff , and exists only for (with value )
Q11. Let . Which of the following is true about differentiability at ?
is not differentiable at
is differentiable at and
is differentiable at and
is differentiable at and exists.
Q12. Let for and . Which of the following statements about at is correct?
(i) is continuous at .
(ii) is differentiable at .
(iii) is continuous at .
(i) and (ii) only
(i), (ii) and (iii) all true
Only (i) is true
(ii) and (iii) only
Q13. For which value of is the function
f(x)=\begin{cases}kx^2+1,&x\le1,\$$2pt]x^2+2x,&x>1,\end{cases}differentiable at ?
No real makes differentiable at
Q14. Consider defined on . Which statement is correct regarding and its inverse ?
is differentiable on and is differentiable on .
is differentiable on and strictly increasing (hence invertible), but is not differentiable at .
is differentiable on and is differentiable at with .
is differentiable on but is not invertible.
Q15. Let be differentiable on and suppose for every rational . Which conclusion must hold?
is constant on the set of rational numbers but may be non-constant on irrationals.
is nonzero at some irrational points, so need not be constant.
is constant on .
must be continuous on (hence identically zero only if continuous).
...and 5 more challenging questions available in the interactive simulator.