Continuity and Differentiability
Definition:
$\quad \quad $ If the graph of a function has no break or jump, then it is said to be continuous function. A function which is not continuous is called a discontinuous function.
Continuity of a Function at a Point :
PYQ2023ContinuityAndDifferentiabilityQ1, PYQ2023ContinuityAndDifferentiabilityQ2
$\quad \quad $ A Function $f(x)$ is said to be continuous at some point $x=a$ of its domain if
$$ \lim _{x \rightarrow a} f(x)=f(a) $$
$$\lim _{x \rightarrow a0} f(x)=\lim _{x \rightarrow a+0} f(x)=f(a)$$
$$ f(a0)=f(a+0)=f(a)$$
$$\lbrace \text{LHL at } x=a \rbrace = \lbrace \text{RHL at } x=a \rbrace = \lbrace \text{value of the function at } x=a \rbrace $$
Continuity from Left and Right :
$\quad \quad $ Function $\mathrm{f}(\mathrm{x})$ is said to be
 $\quad $ Left Continuous at $x=a$ if $$\lim _{x \rightarrow a0} f(x)=f(a) \quad$$
$\quad $ $$f(a0)=f(a)$$
 $\quad $ Right Continuous at $x=a$ if $$\lim _{x \rightarrow a+0} f(x)=f(a)$$
$\quad $ $$f(a+0)=f(a)$$
$\quad \quad $ Thus a function $\mathrm{f}(\mathrm{x})$ is continuous at a point $\mathrm{x}=\mathrm{a}$ if it is left continuous as well as right continuous at $\mathrm{x}=\mathrm{a}$.
Continuity in an Interval :

A function $f(x)$ is continuous in an open interval (a, b) if it is continuous at every point of the interval.

A function $\mathrm{f}(\mathrm{x})$ is continuous in a closed interval [a, b] if it is

continuous in (a, b)

right continuous at $\mathrm{x}=\mathrm{a}$

left continuous at $\mathrm{x}=\mathrm{b}$
Continuous Functions :
$\quad \quad$ A function is said to be continuous function if it is continuous at every point in its domain. Following are examples of some continuous functions:

$ f(x)=x$ (Identify function)

$ f(x)=c$ (Constant function)

$ f(x)=a_{0} x^{n}+a_{1} x^{n1}+\ldots . .+a^{n}$ (Polynomial function)

$ f(x)=\sin x, \cos x$ (Trigonometric function)

$ f(x)=a^{x}, e^{x}, e^{x}$ (Exponential function)

$ f(x)=\log x$ (Logarithmic function)

$ f(x)=\sinh x, \cosh x, \tanh x$ (Hyperbolic function)

$ f(x)=x, x+x, xx, xx$ (Absolute value functions)
Discontinuous Functions :
$\quad \quad $ A function is said to be a discontinuous function if it is discontinuous at at least one point in its domain. Following are examples of some discontinuous functions:
No.  Functions  Points of discontinuity 

(i)  $\lfloor x \rfloor$  Every Integer 
(ii)  $x  \lfloor x \rfloor$  Every Integer 
(iii)  $\frac{1}{x}$  $x = 0$ 
(iv)  $\tan x, \sec x$  $x = \pm\frac{\pi}{2}, \pm\frac{3\pi}{2}, \ldots$ 
(v)  $\cot x, \csc x$  $x = 0, \pi, +2\pi, \ldots$ 
(vi)  $\frac{1}{\sin x}, \frac{1}{\cos x}$  $x = 0$ 
(vii)  $e^{1/x}$  $x = 0$ 
(viii)  $\cot x, \csc x$  $x = 0$ 
Properties of Continuous Functions :

The sum, difference, product, quotient (if $\mathrm{Dr} \neq 0$ ) and composite of two continuous functions are always continuous functions.

if $\mathrm{f}(\mathrm{x})$ and $\mathrm{g}(\mathrm{x})$ are continuous functions then following are also continuous functions:

$f(x)+g(x)$

$ f(x)g(x)$

$f(x) \cdot g(x)$

$ \lambda f(x)$, where $\lambda$ is a constant

$\frac{\mathrm{f}(\mathrm{x})}{\mathrm{g}(\mathrm{x})}$, if $\mathrm{g}(\mathrm{x}) \neq 0$

$\mathrm{f}[\mathrm{g}(\mathrm{x})]$
Important Point :
PYQ2023ContinuityAndDifferentiabilityQ1, PYQ2023ContinuityAndDifferentiabilityQ2, PYQ2023MODQ3

The discontinuity of a function $\mathrm{f}(\mathrm{x})$ at $\mathrm{x}=$ a can arise in two ways

If $\lim _{x \rightarrow a^{}} f(x)$ exist but $\neq f(a)$ or $\lim _{x \rightarrow a^{+}} f(x)$ exist but $\neq f(a)$, then the function $f(x)$ is said to have a removable discontinuty.

The function $\mathrm{f}(\mathrm{x})$ is said to have an unremovable discontinuity when $\lim _{\mathrm{x} \rightarrow \mathrm{a}} \mathrm{f}(\mathrm{x})$ does not exist.

$$ \text { i.e. } \lim _{x \rightarrow a^{}} f(x) \neq \lim _{x \rightarrow a^{+}} f(x) $$
 Differentiability at a point
$\quad \quad$ Let $f(x)$ be a ral valued function defined on an open interval $(a, b)$ and let $c \in(a, b)$. Then $f(x)$ is said to be differentiable or derivable at $x=c$, iff $$\lim _{x \rightarrow c} \frac{f(x)f(c)}{xc} \quad \text{exists finitely} $$
$\quad \quad$ This limit is called the derivative or differential coefficient of the function $\mathrm{f}(\mathrm{x})$ at $\mathrm{x}=\mathrm{c}$, and is denoted by $\mathrm{f}^{\prime}(\mathrm{c})$ or
$$ \begin{gathered} \operatorname{Df}(c) \text { or } \left\lbrace \frac{d}{d x} f(x)\right\rbrace _{x=c} \ \lim _{x \rightarrow c^{}} \frac{f(x)f(c)}{h}=\lim _{h \rightarrow 0} \frac{f(c+h)f(c)}{h} \end{gathered} $$
$\quad \quad$ is called the left hand derivative of $f(x)$ at $x=c$ and is denoted by $f’(c^) \text{ or } \mathrm{Lf}’(c) \text{ while.}$
$$ \lim _{x \rightarrow c^{+}} \frac{f(x)f(c)}{xc} \operatorname{or} \lim _{h \rightarrow 0} \frac{f(c+h)f(c)}{h} $$
$\quad \quad$ is called the righthand derivative of $f(x)$ at $x=c$ and is denoted by $f’(c^+)$ or $Rf f’(c)$.
 LeftHand Derivative (LHD)
$\quad \quad$ If $f(x)$ is defined in some neighborhood of $c$ and the limit $$ \lim _{h \rightarrow 0^} \frac{f(c+h)f(c)}{h} $$ $\quad \quad$ exists, then it is called the lefthand derivative of $f(x)$ at $x = c$ and is denoted by $f’(c^)$ or $Lf f’(c)$.
 RightHand Derivative (RHD)
$\quad \quad$ If $f(x)$ is defined in some neighborhood of $c$ and the limit $$ \lim _{h \rightarrow 0^+} \frac{f(c+h)f(c)}{h} $$ $\quad \quad$ exists, then it is called the righthand derivative of $f(x)$ at $x = c$ and is denoted by $f’(c^+)$ or $Rf f’(c)$.
$\quad \quad$ If $Lf f’(c) \neq Rf f’(c)$, we say that $f(x)$ is not differentiable at $x=c$.
Differentiability in a Set :
$\quad \quad$ A function $f(x)$ defined on an open interval $(a, b)$ is said to be differentiable or derivable in the open interval $(a, b)$ if it is differentiable at each point of $(a, b)$.