Continuity And Differentiability

Continuity

If the Graph of a function has no break or gap, then it is continuous, otherwise it is discontinuous. A function which is not continuous is called a discontinuous function. E.g. Graph of cos $x$ is continuous.

While $f\left(x\right)=\frac{1}{x}$ is discontinuous at $x=0$

Continuity of a Function at a Point

Let $f$ be a real function and $a$ be in the domain of $f.$ We say $f$ is continuous at $a,$if $\underset{x\to a}{\lim }f\left(x\right)=f\left(a\right).$

i.e. $\underset{x\to a^{-}}{\lim }f\left(x\right)=\underset{x\to a^{+}}{\lim }f\left(x\right)=f\left(a\right)$

Hence, $f\left(x\right)$ is continuous, if $\underset{x\to a}{\lim }f\left(x\right)$exists and equals to $f\left(a\right).$

Note If a function is not continuous at $x=c.$ then if is said to be discontinuous at c.

Continuity of a Function in an Interval

1. A function $f\left(x\right)$ is said to be continuous in an open interval $\left(a,b\right),$ if $f\left(x\right)$ is continuous at every point of the interval.
2. A function $f\left(x\right)$is said to be continuous in a closed interval $\left[a,b\right],$ if $f\left(x\right)$ is continuous in $\left(a,b\right).$ Additionally, $f\left(x\right)$ is continuous at $x=a$ from right hand limit and continuous at $x=b$ from left hand limit.

Note A real function $f$ is said to be continuous in its domain, if it is continuous at every point in its domain.

Discontinuity of a Function.

The function $f\left(x\right)$ can be discontinuous at a point $x=a$ in any one of the following ways.

1. $f\left(a\right)$ is not defined.

2. $LHL$ and $RHL$ both exist but unequal i.e.

$\underset{x\to a^{-}}{\lim }f\left(x\right)\ne \underset{x\to a^{+}}{\lim }f\left(x\right)$

3. Either $\underset{x\to a^{-}}{\lim }f\left(x\right)$or $\underset{x\to a^{+}}{\lim }f\left(x\right)$or both non-existing or infinite.

4. $LHL$ and $RHL$ both exist and equal but not equal to $f\left(a\right),$ $\underset{x\to a^{-}}{\lim }f\left(x\right)=\underset{x\to a^{+}}{\lim }f\left(x\right)\ne f\left(a\right)$

Fundamental Theorems of Continuity

1. If $f$ and $g$ are continuous functions, then

a. $f\pm g$ and $fg$ are continuous.

b. $cf$ is continuous, where $c$ is a constant.

c. $\frac{f}{g}$ is continuous at those points where $g\left(x\right)\ne 0.$

2. If is continuous at a point $a$ and $f$ is continuous at $g\left(a\right),$then $fog$ is continuous at $a.$

3. If $f$ is continuous in $\left[a,b\right],$ then it is bounded in $\left[a,b\right]i.e$ there exist $m$ and $M$ such that

$m\le f\left(x\right)\le M,\forall x\in \left[a,b\right],$

Where $m$ and $M$ are called minimum and maximum values of between minimum value of $f\left(x\right)$ respectively in the interval $\left[a,b\right],$

4. For any value of $x\in \left[a,b\right]$ there exists a value of $f\left(x\right)$ which lies between minimum and maximum. Thus,$a\le x\le b\Rightarrow m\le f\left(x\right)\le M$ or range of $f\left(x\right)=\left[m,M\right],$if $x\in \left[a,b\right].$

5. If $f$ is continuous in its domain, then $\left|f\right|$ is also continuous in its domain.

6. If $f$ is continuous at $a$ and $f\left(a\right)\ne 0,$ then there exists an open interval $\left(a-\delta ,a+\delta \right)$ such that for all $x\in \left(a-\delta ,a+\delta \right),f\left(x\right)$ has the same sign as $f\left(a\right).$

7. If $f$ is a continuous function defined on $\left[a,b\right]$ such that $f\left(a\right)$and $f\left(b\right)$ are opposite sign, then there exists at least one solution of the equation $f\left(x\right)$=0 in the open interval $\left(a,b,\right),$

8. If $f$ is continuous in domain $D,$ then $\frac{1}{f}$ is also continuous in $D-\left\{x:f\left(x\right)=0\right\}.$

Differentiability (or Derivability) of a Function at a Point

The function $f\left(x\right)$ is differentiable at a point $P$ iff there exists a unique tangent at point $P$.

In other words, $f\left(x\right)$ is differentiable at a point $P$ iff the curve does not have $P$ as a corner point i.e. the function is not differentiable at those points on which function has holes or sharp edges.

Differentiability

If the curve has no break point and no sharp edge, then it is differentiable.

Mathematically A function $f\left(x\right)$ is said to be differentiable at a point

$a$ in its domain, if $\underset{x\to a}{\lim }\frac{f\left(x\right)-f\left(a\right)}{x-a}$ exist finitely or iff

$\underset{x\to a^{-}}{\lim }\frac{f\left(x\right)-f\left(x\right)}{x-a}=\underset{x\to a^{+}}{\lim }\frac{f\left(x\right)-f\left(a\right)}{x-a}$

i.e. Left Hand Derivative $\left(LHD\right)=$ Right Hand Derivative $\left(RHD\right)$

Differentiability of a Function in an Interval

1. A function $f\left(x\right)$ is said to be differentiable in an interval $\left(a,b\right),$ if $f\left(x\right)$ is differentiable at every point of this interval$\left(a,b\right).$
2. A function $f\left(x\right)$ is said to be differentiable in a closed interval $\left[a,b\right],$if $f\left(x\right)$ is differentiable in $\left(a,b\right),$ in addition $f\left(x\right)$ is differentiable at $x=a$ from right hand derivative and differentiable at $x=a$ from left hand derivative

Relation between continuity and Differentiability

1. If a function $f\left(x\right)$ is differentiable at$x=a,$ then $f\left(x\right)$ is necessarily continuous at $x=a$ but the converse is not necessary true.
2. The sum, difference, product and quotient of two differentiable function is differentiable .The composition of differentiable. Function is a differentiable function.