**Continuous and discontinuous function:**A function f is said to be continuous at a number **“c” if and only if** the following three cases are satisfied:

**Discontinuous function:**

If one or more of these three condition are** fail to hold at “c”** then function is said to be **discontinuous function.**

**Continuous and discontinuous Function:**

** Example:**

Consider the function:

**f(x) = x ² – 1 / x-1 **

or

** (x -1)(x + 1) /x -1)**

because

** a ² – b ² = (a – b) (a + b)**

**Solution:**

Here, f(1) is not defined

**⇒ f(x) is discontinuous at 1.**

Further

**Therefore, f(x) is continuous at any other number whose x ≠ 1.**

**Continuous function example:**

**For f(x) = 3x² – 5x + 4**

discuss continuity of “f”

at x = 1

**SOLUTION:**

**And**

Therefore

f(x) is continuous at x = 1:

**Discontinuous Function Example:**

discuss the continuity of f

** at x =3**

**f(x) = x-1, if x < 3**

**f(x) = 2x + 1, if 3 ≤ 1**

**Solution:**

from the graph, we see that there is a break in the figure at the point when x = 3

Now f(3) = 2(3) + 1 =7

**this implies that(⇒) condition (i) is satisfied**

That is, condition (ii) is not satisfied

**Hence, f(x) is not continuous at x = 3**

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