Real Analysis & Calculus Revision Guide

Complete Real Analysis & Calculus Revision Guide

Continuity • Uniform Continuity • Differentiability • Monotone Functions • Sequences • Limit Points • Topology & Theorems

1. Boundedness Theorem

If a function f is continuous on a closed interval [a,b], then it is bounded.

There exist real numbers M and m such that:

m ≤ f(x) ≤ M for all x ∈ [a,b]

Example

f(x)=x² on [-2,2]

Minimum value = 0

Maximum value = 4

Hence f(x) is bounded.

Continuous functions on closed intervals never "blow up" to infinity.

2. Extreme Value Theorem

If f is continuous on [a,b], then f attains both:

• Absolute Maximum
• Absolute Minimum

Example

f(x)=x² on [-1,2]

• Minimum = 0 at x=0
• Maximum = 4 at x=2

3. Intermediate Value Theorem (IVT)

If f is continuous on [a,b] and k lies between f(a) and f(b), then there exists c∈(a,b) such that:

f(c)=k

Example

f(x)=x³

f(1)=1 and f(2)=8

Since 5 lies between 1 and 8, there exists c∈(1,2) such that:

f(c)=5

4. Zero Existence Theorem

If:

f(a)·f(b) < 0

then there exists c∈(a,b) such that:

f(c)=0

Example

f(x)=x²−2

f(1)=-1

f(2)=2

Sign changes occur, therefore root exists between 1 and 2.

5. Continuity

A function is continuous at x=c if:

lim(x→c) f(x) = f(c)

Examples

Function	Continuity
x²	Continuous everywhere
|x|	Continuous everywhere
1/x	Discontinuous at x=0

6. Piecewise Continuity Example

Function (a)

f(x)=x for x<0
f(0)=0
f(x)=x² for x>0

Left-hand limit = 0

Right-hand limit = 0

Function is continuous everywhere.

Function (b)

f(x)=cos(x) for x<0
f(0)=1/2
f(x)=x² for x>0

LHL = 1

RHL = 0

LHL ≠ RHL, therefore discontinuous at x=0.

7. Uniform Continuity

Uniform continuity means the same δ works everywhere in the domain.

Important Results

Condition	Result
Continuous on [a,b]	Uniformly Continuous
Bounded Derivative	Uniformly Continuous
Continuous Periodic Function	Uniformly Continuous

Examples

Function	Uniformly Continuous?
xsin(1/x)	Yes
sin(1/x)	No
1/x	No
sin(x)	Yes
|x|	Yes
e^(-x²)	Yes

8. Differentiability

f is differentiable at c if:

f'(c)=lim (f(x)-f(c))/(x-c)

Example

f(x)=x²

f'(x)=2x

Differentiable everywhere.

Continuity does not imply differentiability.

f(x)=|x|

Continuous at x=0 but not differentiable at x=0.

9. Left and Right Derivatives

Left Derivative:

Lf'(c)=lim x→c⁻ (f(x)-f(c))/(x-c)

Right Derivative:

Rf'(c)=lim x→c⁺ (f(x)-f(c))/(x-c)

Function is differentiable iff:

Lf'(c)=Rf'(c)

f(x)=|x|

Left derivative = -1

Right derivative = +1

Not differentiable at x=0.

10. Extreme Points

If f has a local maximum or minimum at interior point c and derivative exists, then:

f'(c)=0

Examples

Function	Extremum
x²	Minimum at x=0
-x²	Maximum at x=0
x³	f'(0)=0 but no extremum

11. Monotone Functions

A monotone function can only have jump discontinuities.

• One-sided limits always exist.
• Discontinuities are at most countable.
• No oscillatory discontinuities.

12. Zeros of Continuous Functions

Let:

Z(f)={x : f(x)=0}

For continuous functions, Z(f) is always closed.

f(x)=sin(x)

Z(f)={nπ : n∈ℤ}

13. Dirichlet Function

f(x)=0 if x is rational
f(x)=1 if x is irrational

Discontinuous everywhere.

14. Modified Dirichlet Function

f(x)=0 if x is rational
f(x)=x²−1 if x is irrational

Continuous only at:

x=−1 and x=1

15. Sequences and Limit Points

Exactly k Limit Points

Create k subsequences converging to k different values.

Limit points {0,1,2}

xₙ= 0+1/n, n≡0(mod3)
1+1/n, n≡1(mod3)
2+1/n, n≡2(mod3)

Uncountably Many Limit Points

Possible.

Example: Enumeration of rationals in [0,1].

Limit point set = [0,1]

16. Sequence Whose Limit Point Set Is Entire ℝ

Enumerate all rational numbers in ℝ.

Since ℚ is dense in ℝ, every real number becomes a subsequential limit.

Limit point set = ℝ.

17. Characteristic Function

χE(x)=1 if x∈E
χE(x)=0 if x∉E

χE is continuous iff E is both open and closed (clopen).

18. Distance Function Example

φ(x)=min(|x−1|,|x−2|)

Continuous everywhere.

Not differentiable at:

• x=1
• x=3/2
• x=2

Final Exam Checklist

• Continuous on closed interval ⇒ bounded + attains extrema.
• IVT guarantees intermediate values.
• Bounded derivative ⇒ Uniform continuity.
• Differentiable ⇒ Continuous.
• Continuous does NOT imply differentiable.
• Monotone ⇒ at most countably many discontinuities.
• Zero set of continuous function is closed.
• Dense sequences can have uncountably many limit points.

Uniform Continuity: Important Examples & Counterexamples

Uniform continuity is one of the most important topics in Real Analysis. A function is uniformly continuous if the same δ works for the entire domain. The following examples are frequently asked in examinations and competitive tests.

✅ Uniformly Continuous Functions

Function	Domain	Why Uniformly Continuous?
e-x²	ℝ	Derivative is bounded.
e-x-x²	ℝ	Derivative remains bounded.
e-x²+x	ℝ	Derivative is bounded due to exponential decay.
sin(√x)	[0, ∞)	Continuous and satisfies a Hölder-type estimate.
|x|	ℝ	Lipschitz continuous.
sin(x)	ℝ	Derivative cos(x) is bounded.
x	ℝ	Lipschitz continuous with constant 1.
ex	(-∞, 0)	Derivative ex ≤ 1.
log(x)	(1, ∞)	Derivative 1/x ≤ 1.
x sin(1/x)	(0,1)	Extends continuously at x = 0.

Quick Rule: If a function has a bounded derivative on its domain, then it is uniformly continuous.

❌ Not Uniformly Continuous Functions

Function	Domain	Reason
ex	ℝ	Derivative grows without bound.
ex	(0, ∞)	Unbounded growth.
log(x)	(0, ∞)	Derivative blows up near 0.
log(x)	(0,1)	Derivative 1/x becomes unbounded near 0.
1/x	(0,1)	Becomes unbounded near 0.
sin(1/x)	(0,1)	Oscillates infinitely fast near 0.
tan(πx/2)	(0,1)	Blows up as x → 1.

Common Exam Trick: A function may be continuous everywhere on its domain but still fail to be uniformly continuous because of rapid oscillation (like sin(1/x)) or unbounded growth (like 1/x and e^x).

Summary Table

Uniformly Continuous	Not Uniformly Continuous
e-x²	ex on ℝ
e-x-x²	ex on (0,∞)
e-x²+x	1/x on (0,1)
sin(√x)	sin(1/x) on (0,1)
|x|	log(x) on (0,∞)
sin(x)	log(x) on (0,1)
x	tan(πx/2) on (0,1)
ex on (-∞,0)	-
log(x) on (1,∞)	-
x sin(1/x)	-