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.

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All Chapter Notes, Concept and Important Formula

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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.

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3. 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 .

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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 $f(c)=y_{0}.$

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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