STUDY MATERIAL
CONTINUOUS FUNCTIONS.
INTRODUCTION:
We have noticed that the limit of a function as x approaches
‘a’ can often be found by simply calculating the value of the function at ‘a’. Functions with
this property are called continuous functions. The mathematical notion of continuity is very
closely related to the everyday usage of the word continuity. We see a lot of continuous
processes occurring in nature. For instance the movement of a river along its course or the
motion of planets around the sun are all examples of continuous processes. The path traced
by a ball projected in space is a parabola which is a continuous curve.
DEFINITION:
Let f be a real valued function. Let I be an open interval. Let a be in I.
A function f is continuous at a number ‘ a’ if lim 𝑓(𝑥) = 𝑓(𝑎).
𝑥→𝑎
If f is not continuous at ‘a’ we say f is discontinuous at ‘a’. Hence the definition of
continuous function implies that if f is continuous at ‘a’ then
1. f(a) is defined .(This means ‘a’ is in the domain of f.)
2. lim 𝑓(𝑥) exists.
𝑥→𝑎
3. lim 𝑓(𝑥) = 𝑓(𝑎).
𝑥→𝑎
If a function coincides with a continuous function in an open interval (a,b) then f is
continuous at ‘c’ where a<c<b.
Roughly speaking we can say that if f is continuous at ‘a’, then f(x) gets closer and closer to
f(a) as x gets closer and closer to ‘a’.
Thus continuity at a point ‘a’ depends not only on the value of the function at ‘a’ but also on
the value of the function around ‘a’.
Geometrically we can say that if a function is continuous at a fixed point then we would be
able to draw the graph of the function around the point without lifting the pen from the plane
of the paper.
In contrast a discontinuous function is one which has gaps or holes in it.
EXAMPLE 1.
Let f(x)=sin(x)
.
It can be seen from the graph that sin(x) is a continuous function.
We also know lim 𝑠𝑖𝑛𝑥 = sina.
𝑥→𝑎
EXAMPLE 2.
Consider f(x) = 𝑥 5 + 4𝑥 3 +3x+5.
Let ‘a’ be any point in the domain of f. We have already learnt to find the limit of
polynomials at specific points in the domain we
substitute the value of the point in the function.
. lim 𝑓(𝑥) = 𝑎5 + 4𝑎3 +3a+5 = f(a). Hence f(x) is continuous.
𝑥→𝑎
All polynomial functions can be sketched without lifting the hand.
Hence polynomial functions are in general continuous functions.
EXAMPLE 3. Let f(x) =-1 if x<0
= 0 if x=0
= +1 if x>0
lim 𝑓(𝑥) =1
𝑥→0+
𝑥→0−
lim 𝑓(𝑥) = −1.
lim 𝑓(𝑥) does not exist. Hence f is discontinuous at x=0
𝑥→0
EXAMPLE 4.
Let f(x)=x
x≠1
=2
x=1
lim 𝑓(𝑥) ≠ 𝑓(1)
𝑥→1
In this example we can observe that even though the limit of f(x) exists at x=1 it is not equal
to the value of the function at x=0.
EXAMPLE 5
1
Let f(x) = 𝑥 .
This function is continuous at every point where it is defined.
Notice that it is not defined at x=0.
EXAMPLE 6
1
Let f(x) = sin 𝑥
x≠ 0
= 0
x= 0
1
sin𝑥 is oscillating between -1 and 1 that it never attains a finite value.
Hence lim 𝑓(𝑥) does not exist. Hence f is discontinuous at x=0.
𝑥→0
EXAMPLE 7.
Let f(x) =|𝑥|
This function coincides with the polynomial function y=x for x>0.
Hence it is continuous at all x>0. It coincides with the polynomial function y=-x for x<0.
Hence it is continuous for x<0.
It remains to discuss continuity at x=0.
lim |𝑥| =0.
𝑥→0+
lim |𝑥| =0 and f(0) =0.
𝑥→0−
Hence |x| is continuous at all real numbers.
EXAMPLE 8.
Consider y = [x]. This function has no limits at integer points.
Let ‘a’ be an integer. Then lim [x] = a while
𝑥→𝑎+
lim [x] = a-1. Hence the limit of the function does not exist if ‘a’ is an integer. However for
𝑥→𝑎−
non-integral values of ‘a’ the limit exists. Let ‘a’ be not an integer. Then lim [x] = lim [x]
𝑥→𝑎+
= a. See the graph.
𝑥→𝑎−
EXAMPLE 9.
1
Let f(x) = 𝑥 2
This function is not defined at x=0. Everywhere else it is continuous.
Hence we say that it is continuous at all points in its domain.
Observe its graph.
ALGEBRA OF CONTINUOUS FUNCTIONS:
Theorem: If f and g are real continuous functions at a real number ‘c’
Then
1. f+g is continuous at x=c.
2. f-g is continuous at x=c.
3. f.g is continuous at x=c.
4. f/g is continuous at x=c.(provided g(c) ≠ 0).
5. Composition of continuous functions is continuous.
PROOF:
1.lim (𝑓 + 𝑔)(𝑥) =lim 𝑓(𝑥) +lim 𝑓(𝑥) =f(a) + g(a).
𝑥→𝑎
𝑥→𝑎
𝑥→𝑎
Hence f+g is continuous at x=c.
All the other properties can be proved similarly.
EXAMPLE 9
Every rational function is continuous.
𝑝(𝑥)
Every rational function can be expressed in the form r(x)= 𝑞(𝑥).[q(x)≠0]
Where p(x) and q(x) being polynomial functions are contiuous.
r(x) being a quotient of two polynomial functions is also continuous wherever it is defined.
EXAMPLE 10.
tan(x) is a continuous function.
Both sin(x) and cos(x) are continuous functions. Hence tan(x) is continuous where all
𝜋
cos(x)≠0. We know that cos(x) is zero for all odd multiples of 2 . Excepting for these values
tan(x) is continuous.
EXAMPLE 11.
On what intervals are the following continuous?
a. f(x) = 𝑥100 -2𝑥 50 +1
It is a polynomial function and hence continuous in (-∞,∞).
𝑥+5
b. g(x) =𝑥 2 −1
g is a quotient of two polynomial functions and hence continuous in
in (-∞,-1), (-1,1) and (1,∞).
c. h(x) = [x]
h is a composition of the square root function and the identity function which are both
continuous and composition of continuous functions is continuous.
DIFFERENTIABLITY.
We have studied in class XI that the slope of a curve y=f(x) at the point where x=a to be
lim
ℎ→0
𝑓(𝑎+ℎ)−𝑓(𝑎)
ℎ
.
We called this limit when it existed to be the derivative of f at a.
A function has a derivative at a point a if f the slopes of the secant lines through A(a, f(a))
and a nearby point B on the graph approach a limit as B approaches A.
Whenever the secant fails to take up a limiting position or if it becomes vertical as B tends to
A the derivative does not exist.
A function whose graph is smooth otherwise but has a corner where the one sided derivatives
differ or a cusp where the slope of AB approaches ∞ from one side and approaches -∞ from
the other side will not be differentiable at that point. If a function is discontinuous at a point it
will fail to be differentiable there. Let us see some graphs of functions which have points
where the derivative does not exist.