Summary Continuity and Differentiability - Mathematics

Chapter 5
CONTINUITY AND
DIFFERENTIABILITY
5.1 Overview

5.1.1 Continuity of a function at a point

Let f be a real function on a subset of the real numbers and let c be a point in the
domain of f. Then f is continuous at c if

lim f ( x) = f (c)
x →c


More elaborately, if the left hand limit, right hand limit and the value of the function
at x = c exist and are equal to each other, i.e.,

lim f ( x) = f (c) = lim+ f ( x)
x→c − x→c


then f is said to be continuous at x = c.
5.1.2 Continuity in an interval
(i) f is said to be continuous in an open interval (a, b) if it is continuous at every
point in this interval.
(ii) f is said to be continuous in the closed interval [a, b] if
 f is continuous in (a, b)

 xlim
→a+
f (x) = f (a)

 xlim
→b –
f (x) = f (b)

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5.1.3 Geometrical meaning of continuity
(i) Function f will be continuous at x = c if there is no break in the graph of the
function at the point ( c, f (c ) ) .
(ii) In an interval, function is said to be continuous if there is no break in the
graph of the function in the entire interval.

5.1.4 Discontinuity

The function f will be discontinuous at x = a in any of the following cases :

(i) xlim
→a−
f (x) and xlim
→a+
f (x) exist but are not equal.

(ii) xlim
→a−
f (x) and xlim
→a+
f (x) exist and are equal but not equal to f (a).

(iii) f (a) is not defined.

5.1.5 Continuity of some of the common functions
Function f (x) Interval in which
f is continuous
1. The constant function, i.e. f (x) = c
2. The identity function, i.e. f (x) = x R
3. The polynomial function, i.e.
f (x)= a0 xn + a1 x n–1
+ ... + an–1 x + an
4. | x – a | (– ∞ , ∞ )
5. x–n, n is a positive integer (– ∞ , ∞ ) – {0}
6. p (x) / q (x), where p (x) and q (x) are R – { x : q (x) = 0}
polynomials in x
7. sin x, cos x R

π
8. tan x, sec x R– { (2 n + 1) : n ∈ Z}
2
9. cot x, cosec x R– { (nπ : n ∈ Z}

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10. e x R
11. log x (0, ∞ )
12. The inverse trigonometric functions, In their respective
i.e., sin–1 x, cos–1 x etc. domains

5.1.6 Continuity of composite functions

Let f and g be real valued functions such that (fog) is defined at a. If g is continuous
at a and f is continuous at g (a), then (fog) is continuous at a.

5.1.7 Differentiability
f ( x + h) − f ( x )
The function defined by f ′ (x) = lim , wherever the limit exists, is
h →0 h
defined to be the derivative of f at x. In other words, we say that a function f is
f (c + h ) − f (c )
differentiable at a point c in its domain if both lim− , called left hand
h →0 h
f (c + h ) − f (c )
derivative, denoted by Lf ′ (c), and lim+ , called right hand derivative,
h →0 h
denoted by R f ′ (c), are finite and equal.

(i) The function y = f (x) is said to be differentiable in an open interval (a, b) if
it is differentiable at every point of (a, b)
(ii) The function y = f (x) is said to be differentiable in the closed interval [a, b]
if R f ′ (a) and L f ′ (b) exist and f ′ (x) exists for every point of (a, b).
(iii) Every differentiable function is continuous, but the converse is not true

5.1.8 Algebra of derivatives
If u, v are functions of x, then

d (u ± v) du dv d dv du
(i) = ± (ii) (u v) = u + v
dx dx dx dx dx dx

du dv
v −u
(iii) d  u  = dx 2 dx
dx  v  v

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5.1.9 Chain rule is a rule to differentiate composition of functions. Let f = vou. If
dt dv df dv dt
t = u (x) and both and exist then = .
dx dt dx dt dx

5.1.10 Following are some of the standard derivatives (in appropriate domains)

d 1 d −1
1. dx (sin x) = 2. dx (cos x) =
–1 –1

1 − x2 1 − x2

d 1 d −1
3. (tan –1 x) = 4. (cot –1 x) =
dx 1 + x2 dx 1 + x2

d 1
5. dx (sec x) = , x >1
–1

x x2 − 1

d −1
6. dx (cosec x) = , x >1
–1

x x2 − 1

5.1.11 Exponential and logarithmic functions
(i) The exponential function with positive base b > 1 is the function
y = f (x) = bx. Its domain is R, the set of all real numbers and range is the set
of all positive real numbers. Exponential function with base 10 is called the
common exponential function and with base e is called the natural exponential
function.
(ii) Let b > 1 be a real number. Then we say logarithm of a to base b is x if bx=a,
Logarithm of a to the base b is denoted by logb a. If the base b = 10, we say
it is common logarithm and if b = e, then we say it is natural logarithms. logx
denotes the logarithm function to base e. The domain of logarithm function
is R+, the set of all positive real numbers and the range is the set of all real
numbers.
(iii) The properties of logarithmic function to any base b > 1 are listed below:
1. logb (xy) = logb x + logb y

x
2. logb   = logb x – logb y
 y