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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 \((\mathrm{a}, \mathrm{b}).\)

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