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1. Continuity of a Function:

A function is said to be continuous on the interval $\left[a,b\right]$, if it is continuous at each point in the interval.

2. Continuity of a Function at a Point:

A function $f\left(x\right)$ is said to be continuous at x = a if $\underset{x\to a}{\lim }f\left(x\right)=f\left(a\right)$. Symbolically, f is continuous at x = a, if $f\left(a-h\right)=f\left(a+h\right)=f\left(a\right),h>0$, i.e., (LHL = $f\left(a\right)$ = RHL).

3. Discontinuity of a Function:

A function is discontinuous at a point x = a, if it is not continuous at that point.

4. Reasons for Discontinuity of a Function at a Point:

All discontinuity points are divided into discontinuities of the first and second kind.

The function f x has a discontinuity of the first kind at x = a, if there exists left-hand limit $f\left(x\right)$and right-hand limit $f\left(x\right)$, or these one-sided limits are finite.

Further, there may be the following two options:

(i) The right-hand limit and the left-hand limit are equal to each other. Such point is having removable discontinuity.

(ii) The right-hand limit and the left-hand limit are unequal. In this case, the function $f\left(x\right)$ has a jump discontinuity.

The function $f\left(x\right)$ is said to have a discontinuity of the second kind (or a non-removable or essential discontinuity) at x = a, if at least one of the left hand or right hand or both either does not exist or is infinite.

5. Removable Type of Discontinuities:

In case $\underset{x\to a}{\lim }f\left(x\right)$exists but is not equal to $f\left(a\right)$, the function is said to have a removable discontinuity or discontinuity of the first kind. In this case, we can redefine the function such that $\underset{x\to a}{\lim }f\left(x\right)=f\left(a\right)$, and make it continuous at x = a. Removable type of discontinuity can be further classified as:

(i) Missing point discontinuity:

Where $\underset{x\to a}{\lim }f\left(x\right)$exists but f a is not defined.

(ii) Isolated point discontinuity:

Where $\underset{x\to a}{\lim }f\left(x\right)$and f a exists but $\underset{x\to a}{\lim }f\left(x\right)\ne f\left(a\right)$

6. Non-Removable Type of Discontinuities:

In case $\underset{x\to a}{\lim }f\left(x\right)$does not exist, then it is not possible to make the function continuous by redefining it. Such discontinuity is known as non-removable discontinuity or discontinuity of the 2nd kind. Non-removable type of discontinuity can be further classified as:

(i) Finite type discontinuity: In such type of discontinuity, left-hand limit and right-hand limit at a point exists but are not equal.

(ii) Infinite type discontinuity: In such type of discontinuity, at least one of the limits viz. L H L and R H L is tending to infinity.

(iii) Oscillatory type discontinuity: In such type of discontinuity, the limits oscillate between two finite quantities.

7. Ways to Remove a Removable Discontinuity:

Function f has a removable discontinuity at x = a, if $\underset{x\to a}{\lim }f\left(x\right)=L$ (for some real number L) but f a ≠ L.

We remove the discontinuity at$a$ by defining a new function as follows:

g x = f x, if x ≠ aand g x = Lif x = a

For all x other than a, we see that g x = f x. So, g is continuous.

8. Properties of Continuous Functions:

If the function f and g are continuous at c, then

f + g is continuous at c;

f - g is continuous at c;

f . g is continuous at c;

$\mathrm{}\frac{f}{g}$ is continuous at c, if g c ≠ 0 and is discontinuous at c, if $g\left(c\right)=0$.

If f is continuous at g x 0 and g is continuous at x 0, then f o g is continuous at x 0.

9. Continuity of Composite Functions:

Assume that f is continuous at a and g is continuous at f a, then the composite function h = g o f is continuous at a.

10. Continuity in an Interval:

A function is said to be continuous in an interval, when the function is defined at every point on that interval and undergoes no interruptions, jumps, or breaks.

If some function f x satisfies these criteria from x = a to x = b, we say that f x is continuous on the interval a , b.

11. Intermediate-Value Theorem:

If f x is continuous on a closed interval a , b and c is any number between $f\left(a\right)$ and f b, inclusive, then there is at least one number x on the interval $\left(a,b\right)$ such that f x = c

12. Differentiability of a Function:

A function f is differentiable at x, if lim h → 0 ⁡ f x + h - f x hexists, i.e., the graph of f has a non-vertical tangent line at x , f x.

13. Concept of Tangent/ Normal and its Association with Derivability:

(i) Tangent

A tangent at a point on the curve is a straight line that touches the curve at that point and whose slope is equal to the gradient/derivative of the curve at that point.

(ii) Normal

A normal at a point on the curve is a straight line that intersects the curve at that point and is perpendicular to the tangent at that point.

14. Relationship between Differentiability and Continuity:

All differentiable functions are continuous, but not all continuous functions are differentiable. So, for a function to be differentiable at $x=a$, it is necessary for the function to be continuous at $x=a$.

Similarly, if a function is not continuous at$x=a$, then it is not differentiable at$x=a$.

15. Properties of differentiable functions:

If $f,g:(a,b)\to R$ are differentiable at $c\in (a,b)$ and$k$ is a constant,$k\in R$, then

(i) $kf,f+g,$ and f g are differentiable at c,

(ii) $\left(kf{)}^{\text{'}}\right(c)=k{f}^{\text{'}}(c),$

(iii) $\left(f+g\right)\text{'}\left(c\right)=f\text{'}\left(c\right)+g\text{'}\left(c\right),$

(iv) ${\left(fg\right)}^{\text{'}}\left(c\right)=f^{\text{'}}\left(c\right)g\left(c\right)+f\left(c\right)g^{\text{'}}\left(c\right),$

(v) If $g\left(c\right)\ne 0$, then $\frac{f}{g}$ is differentiable at c.

16. Rolle's Theorem:

Suppose $f\left(x\right)$ is a function that satisfies the following:

(i) f x is continuous on the closed interval $\left[a,b\right]$,

(ii) f x is differentiable on the open interval a , b,

(iii) f a = f b, then there is a number c such that $a<c<b$ and $f^{\text{'}}\left(c\right)=0$, or in other words f x has a critical point in a , b.

17. The Mean Value Theorem:

Suppose f x is a function that satisfies both of the following:

(i) f x is continuous on the closed on the interval a , b,

(ii) f x is differentiable on the open interval a , b,

then, there is a number c such that a < c < band $f^{\text{'}}\left(c\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

$\Rightarrow f\left(b\right)-f\left(a\right)=f^{\text{'}}\left(c\right)\left(b-a\right)$

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