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

Straight Line - Notes, Concept and All Important Formula

STRAIGHT LINE Table Of Contents 1. RELATION BETWEEN CARTESIAN CO-ORDINATE & POLAR CO-ORDINATE SYSTEM If \((x, y)\) are Cartesian co-ordinates of a point \(P\) , then : \(x=r \cos \theta\) , \(y=r \sin \theta\) and \(r=\sqrt{x^{2}+y^{2}}, \quad \theta=\tan ^{-1}\left(\dfrac{y}{x}\right)\) All Chapter Notes, Concept and Important Formula 2. DISTANCE FORMULA AND ITS APPLICATIONS : If \(\mathrm{A}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)\) and \(\mathrm{B}\left(\mathrm{x}_{2}, \mathrm{y}_{2}\right)\) are two points, then \(\mathbf{A B=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}}\) Note : (i) Three given points \(A, B\) and \(C\) are collinear, when sum of any two distances out of \(\mathrm{AB}, \mathrm{BC}, \mathrm{CA}\) is equal to the remaining third otherwise the points will be the vertices of triangle. (ii) Let \(A, B, C \& D\) be the four given points in a plane. Then the quadrilateral will be: (a) Square if \(A B=B C=C D=D...
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

Binomial Theorem - Notes, Concept and All Important Formula

BINOMIAL THEOREM \((x+y)^{n}={ }^{n} C_{0} x^{n}+{ }^{n} C_{1} x^{n-1} y+{ }^{n} C_{2} x^{n-2} y^{2}+\ldots\) \( . .+{ }^{n} C_{r} x^{n-r} y^{r}+\ldots\) \( . .+{ }^{n} C_{n} y^{n}\) \(=\displaystyle \sum_{ r =0}^{ n }{ }^{ n } C _{ r } x ^{ n - r } y ^{ r }\) , where \(n \in N\) . 1. IMPORTANT TERMS IN THE BINOMIAL EXPANSION ARE : (a) General term: The general term or the \((r+1)^{\text {th }}\) term in the expansion of \((x+y)^{n}\) is given by \(T _{ r +1}={ }^{ n } C _{ r } x ^{ n \cdot r } \cdot y ^{r}\) (b) Middle term : The middle term (s) is the expansion of \((x+y)^{n}\) is (are) : (i) If \(n\) is even, there is only one middle term which is given by \(T _{( n +2) / 2}={ }^{ n } C _{ n / 2} \cdot x ^{ n / 2} \cdot y ^{ n / 2}\) (ii) If \(n\) is odd, there are two middle terms which are \(T _{( n +1) / 2}\) & \(T _{[( n +1) / 2]+1}\) (c) Term independent of x : Term independent of \(x\) contains no \(x\) ; Hence find the value of r for which the exponent of \(x\)...
Post a Comment
Read more