Skip to main content

Continuity - Notes, Concept and All Important Formula

CONTINUITY

1. CONTINUOUS FUNCTIONS:

A function \(f(x)\) is said to be continuous at \(x=a\), if \(\displaystyle \lim _{x \rightarrow a} f(x)\) exists and is equal to \(f(\) a). Symbolically \(f(x)\) is continuous at \(x=a\).
If \(\displaystyle \lim _{h \rightarrow 0} f(a-h)=\displaystyle \lim _{h \rightarrow 0} f(a+h)=f(a)=\) finite and fixed quantity \((\mathrm{h}>0)\).
i.e. \(\left.\mathrm{LHL}\right|_{\mathrm{x}=\mathrm{a}}=\left.\mathrm{RHL}\right|_{\mathrm{x}=\mathrm{a}}=\) value of \(\left.f(\mathrm{x})\right|_{\mathrm{x}=\mathrm{a}}=\) finite and fixed quantity.
At isolated points functions are considered to be continuous.


2. CONTINUITY OF THE FUNCTION IN AN INTERVAL:

(a) A function is said to be continuous in \((a, b)\) if \(f\) is continuous at each & every point belonging to \((a, b)\).

(b) A function is said to be continuous in a closed interval \([a, b]\) if :

 \(\circ\,\, \mathrm{f}\) is continuous in the open interval \((a, b)\).

 \(\circ \,\, \mathrm{f}\) is right continuous at 'a' i.e. \(\displaystyle \lim_{x \rightarrow a^{+}} \mathrm{f}(\mathrm{x})=\mathrm{f}(\mathrm{a})=\mathrm{a}\) finite quantity.

 \(\circ\,\, \mathrm{f}\) is left continuous at 'b' i.e. \(\displaystyle \lim_{x \rightarrow h^{-}} \mathrm{f}(\mathrm{x})=\mathrm{f}(\mathrm{b})=\mathrm{a}\) finite quantity.

Note :

(i) All Polynomials, Trigonometrical functions, exponential & Logarithmic functions are continuous in their domains.

(ii) If \(\mathrm{f} \) & \(\mathrm{~g}\) are two functions that are continuous at \(\mathrm{x}=\mathrm{c}\) then the function defined by : \(\mathrm{F}_{1}(\mathrm{x})=\mathrm{f}(\mathrm{x}) \pm \mathrm{g}(\mathrm{x}) ; \mathrm{F}_{2}(\mathrm{x})=\mathrm{K} \mathrm{f}(\mathrm{x}), \mathrm{K}\) any real number, \(\mathrm{F}_{3}(\mathrm{x})=\mathrm{f}(\mathrm{x}) \cdot \mathrm{g}(\mathrm{x})\) are also continuous at \(\mathrm{x}=\mathrm{c}\). Further, if \(g(c)\) is not zero, then \(F_{4}(x)=\dfrac{f(x)}{g(x)}\) is also continuous at \(x=c\).

(iii) If \(f\) and \(g\) are continuous then fog and gof are also continuous.

(iv) If \(f\) and \(g\) are discontinuous at \(x=c\), then \(f+g, f-g, f . g\) may still be continuous.

(v) Sum or difference of a continuous and a discontinuous function is always discontinuous.


3. REASONS OF DISCONTINUITY:

REASONS OF DISCONTINUITY

(a) Limit does not exist i.e. \(\displaystyle \lim_{x \rightarrow a^{-}} f(x) \neq \displaystyle \lim_{x \rightarrow a^{+}} f(x)\)

(b) \(\displaystyle \lim_{x \rightarrow a} f(x) \neq f(a)\) Geometrically, the graph of the function will exhibit a break at \(\mathrm{x}=\mathrm{a}\), if the function is discontinuous at \(\mathrm{x}=\mathrm{a}\). The graph as shown is discontinuous at \(\mathrm{x}=1,2\) and 3 .


4. THE INTERMEDIATE VALUE THEOREM :

Suppose \(\mathrm{f}(\mathrm{x})\) is continuous on an interval \(\mathrm{I}\) and \(\mathrm{a}\) and \(\mathrm{b}\) are any two points of I. Then if \(y_{0}\) is a number between \(f(a)\) and \(f(b)\), their exists a number \(c\) between \(a\) and \(b\) such that \(f(c)=y_{0}.\)

THE INTERMEDIATE VALUE THEOREM

Note that a function \(\mathrm{f}\) which is continuous in \([\mathrm{a}, \mathrm{b}]\) possesses the following property.

If \(\mathrm{f}(\mathrm{a}) \) & \(\mathrm{f}(\mathrm{b})\) posses opposite signs, then there exists atleast one solution of the equation \(\mathrm{f}(\mathrm{x})=0\) in the open interval \((\mathrm{a}, \mathrm{b}).\)

Labels:

Comments

Post a Comment

Popular posts from this blog

Solultion of Triangle - Notes, Concept and All Important Formula

PROPERTIES AND SOLUTIONS OF TRIANGLE 1. SINE FORMULAE : In any triangle \(\mathrm{ABC}\) \(\dfrac{\mathrm{a}}{\sin \mathrm{A}}=\dfrac{\mathrm{b}}{\sin \mathrm{B}}=\dfrac{\mathrm{c}}{\sin \mathrm{C}}=\lambda=\dfrac{\mathrm{abc}}{2 \Delta}=2 \mathrm{R}\) where \(\mathrm{R}\) is circumradius and \(\Delta\) is area of triangle. All Chapter Notes, Concept and Important Formula 2. COSINE FORMULAE : (a) \(\mathrm{\cos A=\dfrac{b^{2}+c^{2}-a^{2}}{2 b c}}\) or \(\mathrm{a^{2}=b^{2}+c^{2}-2 b c \cos A}\) (b) \(\mathrm{\cos B=\dfrac{c^{2}+a^{2}-b^{2}}{2 c a}}\) (c) \(\mathrm{\cos C=\dfrac{a^{2}+b^{2}-c^{2}}{2 a b}}\) 3. PROJECTION FORMULAE : (a) \(b \cos C+c \cos B=a\) (b) \(c \cos A+a \cos C=b\) (c) \(a \cos B+b \cos A=c\) 4. NAPIER'S ANALOGY (TANGENT RULE) : (a) \(\tan \left(\dfrac{\mathrm{B}-\mathrm{C}}{2}\right)=\dfrac{\mathrm{b}-\mathrm{c}}{\mathrm{b}+\mathrm{c}} \cot \dfrac{\mathrm{A}}{2}\) (b) \(\tan \left(\dfrac{\mathrm{C}-\mathrm{A}}{2}\right)=\dfrac{\mathrm{c}-\mathrm{...
Post a Comment
Read more

Parabola - Notes, Concept and All Important Formula

PARABOLA 1. CONIC SECTIONS : A conic section, or conic is the locus of a point which moves in a plane so that its distance from a fixed point is in a constant ratio to its perpendicular distance from a fixed straight line. (a) The fixed point is called the FOCUS. (b) The fixed straight line is called the DIRECTRIX. (c) The constant ratio is called the ECCENTRICITY denoted by e. (d) The line passing through the focus & perpendicular to the directrix is called the AXIS. (e) A point of intersection of a conic with its axis is called a VERTEX. All Chapter Notes, Concept and Important Formula 2. GENERAL EQUATION OF A CONIC : FOCAL DIRECTRIX PROPERTY: The general equation of a conic with focus \((p, q) \) & directrix \(\mathrm{lx}+\mathrm{my}+\mathrm{n}=0\)   is :  \(\left(1^{2}+\mathrm{m}^{2}\right)\left[(x-\mathrm{p})^{2}+(\mathrm{y}-\mathrm{q})^{2}\right] \) \(=e^{2}(\mathrm{l} \mathrm{x}+\mathrm{m} y+\mathrm{n})^{2}  \) \(\equiv \mathrm{ax}^{2}+2 \mathrm{~h} ...
Post a Comment
Read more

Indefinite Integration - Notes, Concept and All Important Formula

INDEFINITE INTEGRATION If  f & F are function of \(x\) such that \(F^{\prime}(x)\) \(=f(x)\) then the function \(F\) is called a PRIMITIVE OR ANTIDERIVATIVE OR INTEGRAL of \(\mathrm{f}(\mathrm{x})\) w.r.t. \(\mathrm{x}\) and is written symbolically as \(\displaystyle \int \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}\) \(=\mathrm{F}(\mathrm{x})+\mathrm{c} \Leftrightarrow \dfrac{\mathrm{d}}{\mathrm{dx}}\{\mathrm{F}(\mathrm{x})+\mathrm{c}\}\) \(=\mathrm{f}(\mathrm{x})\) , where \(\mathrm{c}\) is called the constant of integration. Note : If \(\displaystyle \int f(x) d x\) \(=F(x)+c\) , then \(\displaystyle \int f(a x+b) d x\) \(=\dfrac{F(a x+b)}{a}+c, a \neq 0\) All Chapter Notes, Concept and Important Formula 1. STANDARD RESULTS : (i) \( \displaystyle \int(a x+b)^{n} d x\) \(=\dfrac{(a x+b)^{n+1}}{a(n+1)}+c ; n \neq-1\) (ii) \(\displaystyle \int \dfrac{d x}{a x+b}\) \(=\dfrac{1}{a} \ln|a x+b|+c\) (iii) \(\displaystyle \int e^{\mathrm{ax}+b} \mathrm{dx}\) \(=\dfrac{1}{...
Post a Comment
Read more