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IC Math 1 Tutorial 2

This tutorial sheet contains a series of mathematical problems focused on limits, continuity, and discontinuity of various functions. It includes proofs, evaluations of limits, and examples of functions that demonstrate specific properties of continuity. The problems are designed for students to apply concepts in real analysis and calculus.

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13 views2 pages

IC Math 1 Tutorial 2

This tutorial sheet contains a series of mathematical problems focused on limits, continuity, and discontinuity of various functions. It includes proofs, evaluations of limits, and examples of functions that demonstrate specific properties of continuity. The problems are designed for students to apply concepts in real analysis and calculus.

Uploaded by

Paladin
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Math-1

(Odd Semester)
Tutorial Sheet-2
Instructor: Prof. Rajendra K. Ray

1. Show that if lim f (x) exists, then it is unique.


x→x0

2. Consider the function f (x) = 5x + 3 with domain A = R. Show that

lim f (x) = 13
x→2

x+1
3. Consider the function f (x) = with domain A = R. Show that
x2 + 3
1
lim f (x) =
x→1 2

1
4. By using the sequential criterion for continuity, prove that the function f (x) = 7 is not continuous at
x
x=0

5. Find lim f (x) for the following functions:


x→0

(a) f (x) = x + 5
(b) f (x) = |x|
x
 ; x ̸= 0
(c) f (x) = sgn(x) = |x|
0; x = 0

6. Determine the points of continuity of



 1 ; x ̸= 0
f (x) = x
0; x = 0

7. Consider the Dirichlet’s function (


1; x ∈ Q
f (x) =
0; x ∈ R/ Q
Show that f is discontinuous everywhere.

8. Prove that a polynomial function is always continuous on R.

9. Consider the function (


2x2 + 3; x ≤ 3
f (x) =
3x + A; x > 3
find the value of A such that f (x) is continuous at x = 3.

10. Prove that the (six basic) trigonometric functions are continuous on their domain.

11. Find the points of discontinuity for the function f (x) = [x], where [x] denotes the greatest integer
function.

1
12. Let I := [a, b] be a closed bounded interval and let f : I → R be continuous on I. Then prove that f is
bounded on I.

13. Give an example of functions f and g that are both discontinuous at a point c in R such that (a) the sum
f + g is continuous at c, (b) the product f g is continuous at c.
x−2
14. Find the point of discontinuity for the function f (x) = .
x2 + 3x − 10
15. Show that a function f (x) = x2 is continuous but not bounded on [0, ∞).

16. Give an example of a function that is not continuous but bounded on a closed interval.

17. Show that the following function is continuous at x = 0 using the ε − δ definition.
(
x2 Sin(1/x); x ̸= 0
f (x) =
0; x = 0

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