Application of derivatives NCERT Chapter 6 |
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Continuity & Differentiability| Class 12| 100 most important questions| CBSE |ISC|JEE-Main | NDA2023
Suppose f is 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 x approaches c = f(c).
More elaborately, if the left hand limit, right hand limit and the value of the function at x = c exist and equal to each other, then f is said to be continuous at x = c. Recall that if the right hand and left hand limits at x = c coincide, then we say that the common value is the limit of the function at x = c. Hence we may also rephrase the definition of continuity as follows: a function is continuous at x = c if the function is defined at x = c and if the value of the function at x = c equals the limit of the function at x = c. If f is not continuous at c, we say f is discontinuous at c and c is called a point of discontinuity of f.
Check the continuity of the function f given by f(x) = 2x + 3 at x = 1.
Examine whether the function f given by f(x) = x2 is continuous at x = 0.
Discuss the continuity of the function f given by f(x) = | x | at x = 0.
Theorem: A real function f is said to be continuous if it is continuous at every point in the domain of f.
Is the function defined by f(x) = | x |, a continuous function?
Theorem 1:
Suppose f and g be two real functions continuous 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).
Differentiate f(x) with respect to x to mean find f ′(x). The following rules were established as a part of algebra of derivatives:
(1) (u ± v)′ = u′ ± v′
(2) (uv)′ = u′v + uv′ (Leibnitz or product rule)
(3) (u/v)^'= (u^' v-uv')/v^2 wherever v ≠ 0 (Quotient rule).
A function is said to be differentiable in an interval [a, b] if it is differentiable at every point of [a, b].
As in case of continuity, at the end points a and b, we take the right hand limit and left hand limit, which are nothing but left hand derivative and right hand derivative of the function at a and b respectively.
Similarly, a function is said to be differentiable in an interval (a, b) if it is differentiable at every point of (a, b).
Theorem 3: If a function f is differentiable at a point c, then it is also continuous at that point.
Derivatives of inverse trigonometric functions
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