Continuity and Differentiability
Definition of continuity at a point
A function f(x) is said to be continuous at x=a if given , there exists such that | f(x) – f(a) | such that | x –a | < .
Alternative definition: f(x) is said to be continuous at x =a if value of f(x) = limit of f(x) at x = a
Or if lim f(x) = f(a)
Of if lim f (a+h) = lim f(a –h) = f(a) …….Equation 1
If the condition (1) is not satisfied, then f(x) is said to be discontinuous at x =a.
Note: In order to rest continuity of a function at a point, we verify the equation (1). This is the working rule of continuity.
Definition of continuity in an interval
Let ‘h’ be a small positive number. A function f(x) is said to be continuous in closed interval [a,b], i.e., if:
(i) f(x) is continuous in open interval (a ,b), i.e.,
(ii) f(x) is continuous at x = 1 from right
(iii) f(x) is continuous at x=b from left
Kinds of discontinuity
(1) Removable Discontinuity: A function f(x) is said to have removable discontinuity at x=a if exists or but . In this case value of function and limit of function are not equal.
(2) Discontinuity of First Kind: If both exist and f (a -0) f (a +0), then f(x) is said to have discontinuity of first kind or ordinary discontinuity at x=a.
(3) Discontinuity of second kind: A function f (x) is said to have discontinuity of second kind at x = a if f(a-0) or f (a + 0 ) or both do not exist.
Properties of continuous function
(1) if f is continuous on a closed interval [a,b], then it is bounded in this interval.
(2) if f is continuous in [a,b] and f(a) and f(b) have opposite signs, then there is at lest one value of x=c such that f(c) = 0 and a<c<b.
(3) If a function f is continuous in closed interval [a,b], then f(x) takes at least once all values of between f(a) and f(b).
Important note: If there is no gap in the graph of the function in a certain range [a,b], then f(x) is said to be continuous in [a,b].
Some standard continuous functions
(1) Every constant function is everywhere continuous.
(2) The identity function I (x) = x is everywhere continuous.
(3) The modulus function f(x) = |x| is continuous .
(4) The exponential function f(x) = and a >0 is continuous everywhere.
(5) The logarithmic function is continuous
(6) Every polynomial function:
A function f(x) is said to be differentiable at x = a if exists and is finite. Let ‘h’ be a small positive number.
Right hand derivative of f(x) at x = a is denoted by Rf’ (a) and is defined as Rf’ (a) =
Left hand derivative of f(x) at x = a is denoted by Lf’ (a) and is defined as Lf’ (a)=
Alternate definition: A function is said to be differentiable at x = a if Rf’ (a) and Lf’ (a) both are finite and Rf’ (a) = Lf’ (a).
The common value is denoted by f’ (a).
Relation between continuity and differentiability
(1) if f(x) is differential at x = a, then it is continuous at x =a.
(2) if f(x) is continuous at x = a, then there is no guarantee that f (x) is differentiable at x=a.
(3) If f (x) is not differentiable at x =a, then it may or may not be continuous at x=a.
(4) if f(x) is not continuous at x=a, then it is not differentiable at x=a.
Some important results on differentiability
(1) Every polynomial function, constant function, exponential function is differentiable .
(2) if f(x) and g(x) are differentiable, then are differentiable functions provided in the last case g(x) .
(3) The composition of differentiable function is differentiable function.
If f(x) is differentiable in the closed interval [a,b] anf f’ (a) and f’ (b) are of opposite signs, then there is a point , ie., a < c < b such that f’ (c) = 0.
Graphical Meaning of Differentiability
F’(a) represents slope of tangent at x=a. A function f(x) is said to be differentiable at x=a if it (tangent) has unique slope at x = a.
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