1. The document discusses continuity of functions, including defining a continuous function as one whose graph can be traced without lifting the pencil, and defining the three conditions for a function f(x) to be continuous at a point a: f(a) must be defined, the limit of f(x) as x approaches a must exist, and the limit must equal f(a). 2. It covers types of discontinuities such as removable, jump, and infinite discontinuities. 3. Theorems are presented stating that arithmetic operations (addition, subtraction, multiplication, division) of continuous functions yield a continuous result. 4. Elementary functions like polynomials, rational fractions, and trigon

1. Continued
Functions
of One
Variable

2. LESSON 1
CONTINUITY OF
FUNCTIONS

3. CONTINUOUS
FUNCTION
It is a function whose graph shows no gaps,
but is continuous line or curve.
They have the property that their graphs can
be traced with a pencil without lifting the
pencil from the page.

4. f(a) is defined
lim f(x) exists
(x→a)
A function f(x) is continuous at
a point a if and only if the
following three conditions are
satisfied.
lim f(x) = f(a)
(x→a)
A function is
discontinuous at a
point a if it fails to be
continuous at a.

5. 1. Check to see if f(a) is defined. If f(a) is undefined, we need go no
further. The function is not continuous at a. If f(a) is defined,
continue to step 2.
2. Compute lim_{x→a}f(x). In some cases, we may need to do this by
first computing lim_{x→a^-}f(x) and lim_{x→a^+}f(x). If
lim_{x→a}f(x) does not exist (that us, it is not real number), then
the function is not continuous at a and the problem is solved. If
lim_{x→a}f(x) exists, then continue to step 3.
3. Compare f(a) and lim_{x→a}f(x). If lim_{x→a}f(x)≠f(a), then the
function is not continuous at a. If lim_{x→a}f(x)=f(a), then the function
is continuous at a.
Problem-Solving Strategy:
Determining Continuity at a Point

6. f(x)=x2+2x+3 at x=0
Example No. 1
a. f(a)
f(0)= (0)2=2(0)+3
f(0)=3
b. lim f(x)
x→a
lim (x2=2x=3)
x→0
=02+2(0)=3
=3
c. lim f(x) = f(a)
x→a
CONTINUOUS!

7. f(x)=x3-1 at x=1
Example No. 2
a. f(a)
f(1)= 13-1
f(1)= 1-1
f(1)=0
b. lim f(x)
x→a
lim (x3-1)
x→1
=13-1
=1-1
=0
c. lim f(x) = f(a)
x→a
CONTINUOUS!

8. g(x)=x2-4/x-2 at x=2
Example No. 3
a. g(a)
g(2)= 22-4/2-2
g(2)=0/0
b. lim g(x)
x→a
lim (x2-4/x-2)
x→2
lim (x+2)(x-2)/x-2
x→2
=2+2
=4
c. lim f(x) ≠ g(a)
x→a
DISCONTINUOUS!

9. LESSON 2
Classification
of breaks
Types of
Discontinuity

10. -this discontinuity occurs when there
is a hole in the graph of the function.
REMOVABLE
DISCONTINUITY
-redefine the function to remove the
discontinuity.
-indeterminate

11. -this discontinuity occurs when the
graph of the function stops at one
point and seems to jump to another
point.
JUMP
DISCONTINUITY
-both left and right hand limit exists
but not equal.

12. -in this type of discontinuity, at least
one of the two limits is infinite.
ASYMPTOTIC/
INFINITE
DISCONTINUITY
-has a vertical asymptote
-rational, logarithmic, and
trigonometric

13. Determine whether the following
functions are continuous or not. If
they are not, classify them as
removable, jump, or infinite.
The discontinuity of the function is removable.
To remove the discontinuity,
redefine the function.

14. Determine whether the following
functions are continuous or not. If
they are not, classify them as
removable, jump, or infinite.
Jump Discontinuity

15. Determine whether the following
functions are continuous or not. If
they are not, classify them as
removable, jump, or infinite.
Infinite Discontinuity

16. LESSON 3
ARITHMETIC
OPERATION ON
CONTINUED
FUNCTION

17. ADDITION OF
CONTINUOUS FUNCTION
THEOREM: Let us say, f and g are two real functions that are
continuous at a point ‘a’, where ‘a’ is a real number. Then the
addition of the two functions f and g is also continuous at ‘a’.
f(x) + g(x) is continuous at x = a
PROOF: Given,
limx→a f(x) = f(a)
limx→ a g(x) = g(a)
Now as per the theorem,
limx → a (f+g)(x) ⇒ lim x → c [f(x) + g(x)]
⇒ limx → c f(x) + limx → c g(x)
⇒ f(a) + g(a)
⇒ (f + g)(a)
Therefore,
limx → a (f+g)(x) = (f + g)(c)
Hence, f+g is continuous at x = a.

18. SUBTRACTION OF
CONTINUOUS FUNCTION
THEOREM: Let us say, f and g are two real functions that are
continuous at a point ‘a’, where ‘a’ is a real number. Then the
subtraction of the two functions f and g is also continuous at ‘a’.
f(x) - g(x) is continuous at x = a
PROOF: Given,
limx→a f(x) = f(a)
limx→ a g(x) = g(a)
Now as per the theorem,
limx → a (f – g)(x) ⇒ lim x → c [f(x) – g(x)]
⇒ limx → c f(x) – limx → c g(x)
⇒ f(a) – g(a)
⇒ (f – g)(a)
Therefore,
limx → a (f – g)(x) = (f – g)(c)
Hence, f – g is continuous at x = a.

19. MULTIPLICATION OF
CONTINUOUS FUNCTION
THEOREM: If f and g are two real functions that are continuous
at a point ‘a’, where ‘a’ is a real number. Then the product of the
two functions f and g is also continuous at ‘a’.
f(x) ● g(x) is continuous at x = a
PROOF: Given,
limx→a f(x) = f(a)
limx→ a g(x) = g(a)
So, the limit of product of two functions, f and g at x is given by:
limx → a (f . g)(x) ⇒ lim x → c [f(x) . g(x)]
⇒ limx → c f(x) . limx → c g(x)
⇒ f(a) . g(a)
⇒ (f . g)(a)
Therefore,
limx → a (f . g)(x) = (f . g)(c)
Hence, f . g is continuous at x = a.

20. DIVISION OF
CONTINUOUS FUNCTION
THEOREM: Suppose, f and g are two real functions that are
continuous at a point ‘a’, where ‘a’ is a real number. Then the
division of the two functions f and g will remain continuous at ‘a’.
f(x) ÷ g(x) is continuous at x = a
Proof: Given,
limx→a f(x) = f(a)
limx→ a g(x) = g(a)
Now as per the theorem,
limx → a (f+g)(x) ⇒ lim x → c [f(x) ÷ g(x)]
⇒ limx → c f(x) ÷ limx → c g(x)
⇒ f(a) ÷ g(a)
⇒ (f ÷ g)(a)
Therefore,
limx → a (f ÷ g)(x) = (f ÷ g)(c)
Hence, f ÷ g is continuous at x = a.

21. LESSON 4
CONTINUITY
OF THE
ELEMENTARY
FUNCTION

22. -These are functions built up of a finite
combination of constant functions, field
operations (addition, multiplication,
division, and root extractions―the
elementary operations) under repeated
compositions. The set of basic
elementary functions includes:
ELEMENTARY
FUNCTIONS
Algebraic Polynomials
Axn+Bxn-1+…+Kx+L
Rational Fractions
Axn+Bxn-1+…+Kx+L
Mxm+Nxm-1+…+Tx+U
Power Functions
xp
Exponential Functions
ax
Logarithmic Functions
logax
Trigonometric Functions
Sin x, cos x, tan x, cot x, sec x, csc x
Inverse Trigonometric Functions
arcsin x, arccos x, arctan x, arccot x,
arcsec x, arccsc x
Hyperbolic Functions
sinh x, cosh x, tanh x, coth x, sech x,
csch x
Inverse Trigonometric Functions
arcsinh x, arccosh x, arctanh x,
arccoth x, arcsechx, arccsch x

23. 2
CONTINUITY OF
ELEMENTARY
FUNCTIONS
All elementary functions are continuous
at any point where they are defined.
EXAMPLE 1
Using the Heine definition, prove that
the function
f(x)=x2
is continuous at any point x=a.
EXAMPLE 2
Using the Heine definition, prove that
the function
f(x)= sec x
is continuous for any x in its domain.

24. 2
EXAMPLE 1
Using the Heine definition, prove that the function
f(x)=x2
is continuous at any point x=a.
Using the Heine definition, we can write the condition of continuity as follows:
where x and y are small numbers shown in the figure.
Using the Heine definition, we can write the condition of continuity as follows:
So that

25. 2
EXAMPLE 2
Using the Heine definition, prove that the function
f(x)=sec x
is continuous for any x in its domain..
The secant function f(x)=sec x=1/cos x has domain all real
numbers x except those of the form
where cosine is zero.
Let x be a differential of independent variable x. Find the corresponding differential function of y.
Calculate the limit as x→0
This result is valid for all x except the roots of the cosine function:
Hence, the range of continuity and the domain of the function f(x)=sec x fully coincide..

26. LESSON 5
PROPERTIES OF
THE CONTINUED
FUNCTION

27. PROPERTIES OF
CONTINUITY