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

Logarithm - Notes, Concept and All Important Formula

LOGARITHM LOGARITHM OF A NUMBER : The logarithm of the number \(\mathrm{N}\) to the base ' \(\mathrm{a}\) ' is the exponent indicating the power to which the base 'a' must be raised to obtain the number \(\mathrm{N}\) . This number is designated as \(\log _{\mathrm{a}} \mathrm{N}\) . (a) \(\log _{a} \mathrm{~N}=\mathrm{x}\) , read as \(\log\) of \(\mathrm{N}\) to the base \(\mathrm{a} \Leftrightarrow \mathrm{a}^{\mathrm{x}}=\mathrm{N}\) . If \(a=10\) then we write \(\log N\) or \(\log _{10} \mathrm{~N}\) and if \(\mathrm{a}=e\) we write \(\ln N\) or \(\log _{e} \mathrm{~N}\) (Natural log) (b) Necessary conditions : \(\mathrm{N}> \,\,0 ; \,\, \mathrm{a}> \,\,0 ; \,\, \mathrm{a} \neq 1\) (c) \(\log _{a} 1=0\) (d) \(\log _{a} a=1\) (e) \(\log _{1 / a} a=-1\) (f) \(\log _{a}(x . y)=\log _{a} x+\log _{a} y ; \,\, x, y> \,\,0\) (g) \(\log _{a}\left(\dfrac{\mathrm{x}}{y}\right)=\log _{\mathrm{a}} \mathrm{x}-\log _{\mathrm{a}} \mathrm{y} ; \,\, \mathrm{...
Post a Comment
Read more

Determinant - Notes, Concept and All Important Formula

DETERMINANT 1. MINORS : The minor of a given element of determinant is the determinant of the elements which remain after deleting the row & the column in which the given element stands. For example, the minor of \(a _{1}\) in \(\left|\begin{array}{lll} a _{1} & b _{1} & c _{1} \\ a _{2} & b _{2} & c _{2} \\ a _{3} & b _{3} & c _{3}\end{array}\right|\) is \(\left|\begin{array}{ll} b _{2} & c _{2} \\ b _{3} & c _{3}\end{array}\right| \&\) the minor of \(b_{2}\) is \(\quad\left|\begin{array}{ll}a_{1} & c_{1} \\ a_{3} & c_{3}\end{array}\right|\) . Hence a determinant of order three will have " 9 minors". 2. COFACTORS : If \(M _{ ij }\) represents the minor of the element belonging to \(i ^{\text {th }}\) row and \(j ^{\text {th }}\) column then the cofactor of that element : \(C_{i j}=(-1)^{i+j} \cdot M_{i j}\) Important Note : Consider \(\Delta=\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} ...
Post a Comment
Read more

Trigonometry Ratios and Identities - Notes, Concept and All Important Formula

TRIGONOMETRIC RATIOS & IDENTITIES Table Of Contents 1. RELATION BETWEEN SYSTEM OF MEASUREMENT OF ANGLES : \(\dfrac{D}{90}=\dfrac{G}{100}=\dfrac{2 C}{\pi}\) 1 Radian \(=\dfrac{180}{\pi}\) degree \(\approx 57^{\circ} 17^{\prime} 15^{\prime \prime}\) (approximately) 1 degree \(=\dfrac{\pi}{180}\) radian \(\approx 0.0175\) radian All Chapter Notes, Concept and Important Formula 2. BASIC TRIGONOMETRIC IDENTITIES : (a) \(\sin ^{2} \theta+\cos ^{2} \theta=1\) or \(\sin ^{2} \theta=1-\cos ^{2} \theta\) or \(\cos ^{2} \theta=1-\sin ^{2} \theta\) (b) \(\sec ^{2} \theta-\tan ^{2} \theta=1\) or \(\sec ^{2} \theta=1+\tan ^{2} \theta\) or \(\tan ^{2} \theta=\sec ^{2} \theta-1\) (c) If \(\sec \theta+\tan \theta\) \(=\mathrm{k} \Rightarrow \sec \theta-\tan \theta\) \(=\dfrac{1}{\mathrm{k}} \Rightarrow 2 \sec \theta\) \(=\mathrm{k}+\dfrac{1}{\mathrm{k}}\) (d) \(\operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1\) or \(\operatorname{cosec}^{2} \theta=1+\cot ^{2} \th...
Post a Comment
Read more