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**A.** **Definition of Continuity **

**Continuity at a Point: **A function f is **continuous at **c if the following three conditions are met.**(i)** f(x) is defined.**(ii)**

**(iii)**

In other words function f(x) is said to be continuous at x = c , if

Symbolically f is continuous at ^{ }x = c

**One-sided Continuity: **

- A function f defined in some neighbourhood of a point c for c ⇒ c is said to be continuous at c from the left if
- A function f defined in some neigbourhood of a point c for x ³ c is said to be continuous at c from the right if
- One-sided continuity is a collective term for functions continuous from the left or from the right.
- If the function f is continuous at c, then it is continuous at c from the left and from the right . Conversely, if the function f is continuous at c from the left and from the right, then

exists & - The last equality means that f is continuous at c.
- If one of the one-sided limits does not exist, then does not exist either. In this case, the point c is a discontinuity in the function, since the continuity condition is not met.

**Continuity In An Interval****(a)** A function f is said to be continuous in an open interval (a , b) if f is continuous at each & every point ∈(a, b).**(b)** A function f is said to be continuous in a closed interval [a,b] if:

- f is continuous in the open interval (a , b) &
- f is right continuous at
**`a'**i.e.

- f is left continuous at
**`b'**i.e.

A function f can be discontinuous due to any of the following three reasons :

- f(x) is not defined at x= c
- Geometrically, the graph of the function will exhibit a break at x= c.

**Example ****1. Test the following functions for continuity****(a) 2x ^{5} - 8x^{2} + 11 / x^{4} + 4x^{3} + 8x^{2} + 8x +4**

Hence f(x) is continuous throughout the entire real line.

**Find A and B so as to make the function continuous.****Solution****. **At x = - π/2

- π/2 - h

where h→ 0

Replace x by - π/2+h

where h → 0

So B - A = 2 ...(i)

At x = π/2

Replace x by π/2 - h

Replace x by π/2+h

where h→ 0

So A+B = 0 ...(ii)

Solving (i) & (ii), B= 1, A = -1**Example ****3. Test the continuity of f(x) at x = 0 if **

**Solution****. **For x < 0,

L.H.L. = R.H.L. ≠ f(0) Hence f(x) is discontinuous at x = 0.**Example ****4. If f(x) be continuous function for all real values of x and satisfies;****x ^{2} + {f(x) – 2} x + 2√3 – 3 – √3 . f(x) = 0, for x ∈ R. Then find the value of f(√3 ).**

Thus,

where

f(x) = x

f(√3) = 2(1-√3).**Example ****5. ****If f (x) is continuous at x = 0, then find the value of (b+c)^{3}-3a.**

N^{r }→ 1 + a + b D^{r }→ 0

for existence of limit a + b + 1 = 0

limit of N^{r} ⇒ 2a+8b = 0 ⇒ a = -4b

hence

-4b+b = -1

⇒ b = 1/3 and a = -4/3

= 8 sin^{2}x - 2 sin^{2}2x / 3x^{4} = 8 sin^{2}x - 8sin^{2}xcos^{2}x / 3x^{4}

= 8 / 3 . sin^{2}x / x^{2} . sin^{2}x / x^{2} = 8 / 3

⇒ e^{A} = 1 / 2 (e^{2x} A / x + B / x) ⇒ x . e^{A} = 1 / 2 (e^{2x} . A + B)**Example ****6. **

**If f is continuous at x = 0, then find the values of a, b, c & d.****Solution. **

,

for existence of limit a + b + 5 = 0

= a / 2 + 5 / 2 - a = 3

⇒ a = - 1 ⇒ b = - 4

for existence of limit c = 0

= ed = 3 ⇒ d = ln 3**Example ****7. Let f(x) = x ^{3} = 3x^{2}_{ + 6 ∀ x ∈ R and}**

x = 0, 2

f"(x) = 6x – 6

f" (0) = –6 < 0 (local maxima at x = 0)

f" (2) = 6 > 0 (local minima at x = 2)

x^{3} – 3x^{2} + 6 = 0 has maximum 2 positive and 1 negative real roots. f(0) = 6.

Now graph of f(x) is :

Clearly f(x) is increasing in (– ∝, 0) U (2, ∝) and decreasing in (0, 2)

⇒ x + 2 < 0 ⇒ x < – 2 ⇒ –3 ≤ x < – 2

⇒ –2 ≤ x + 1 < –1 and –1 ≤ x + 2 < 0

in both cases f(x) increases (maximum) of g(x) = f(x + 2)

g(x) = f(x + 2); –3 ≤ x < – 2 ...(1)

and if x + 1 < 0 and 0 ≤ x + 2 < 2

– 2 ≤ x < –1 then g(x) = f(0)

Now for x + 1 ≥ 0 and x + 2 < 2 ⇒ –1 ≤ x < 0, g(x) = f(x + 1)

Hence g(x) is continuous in the interval [–3, 1].**Example ****8. Given the function, ****f(x) = x [ 1 / x(1 + x) + 1 / (1 + x)(1 + 2x) + 1 / (1 + 2x)(1 + 3x) + ....upto ∞**

**Find f ^{ }(0) if f^{ }(x) is continuous at x = 0.**

f(x) = 2 / 1 + x - 1 / 1 + nx** **upto n terms when x ≠0.**Hence ****Example ****9. Let f: R →R be a function which satisfies f(x+y ^{3}) = f(x) + (f(y))^{3} ∀ x, y ∈ R. **

To prove

Put x = y = 0 in the given relation f(0) = f(0) + (f(0))

Since f is continuous at x = 0

To prove

= f(x) + 0 = f(x).

Hence f is continuous for all x ∈ R.

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