Examine the continuity of the function at x = 0. Also sketch the graph of this function.

__Ideas required to solve the problem:__

1. Meaning of continuity of function – If we talk about a general meaning of continuity of a function f(x) , we can say that if we plot the coordinates (x , f(x)) and try to join all those points in the specified region, we can do so without picking our pen i.e you will put your pen/pencil on graph paper and you can draw the curve without any breakage.

Mathematically we define the same thing as given below:

A function f(x) is said to be continuous at x = c where c is x–coordinate of the point at which continuity is to be checked

If:–

equation 1

where h is a very small positive no (can assume h = 0.00000000001 like this )

It means :–

Limiting value of the left neighbourhood of x = c also called left hand limit LHL must be equal to limiting value of right neighbourhood of x= c called right hand limit RHL and both must be equal to the value of f(x) at x=c i.e. f(c).

Thus, it is the necessary condition for a function to be continuous

So, whenever we check continuity we try to check above equality if it holds true, function is continuous else it is discontinuous.

Let’s Solve now:

Given function is

…… Equation 2

We need to check whether f(x) is continuous at x=0 or not

For this we need to check LHL, RHL and value of function at x=0

Clearly,

f(0) = 3*0 – 2 = –2 [from equation 2]

LHL =

RHL =

As, LHL ≠ RHL

∴ f(x) is discontinuous at x = 0

This can also be proved by plotting f(x) on cartesian plane.

For x >0 ,we need to plot

y = x + 1

put y=0, we get x=–1 and for second point we put x=0 and thus get y=1

two points are enough to plot the straight line.

Two coordinates are (–1,0) and (0,1)

For x≤0, we need to plot

y = 3x – 2

put x = 0 then y = –2

on putting y=0 we get x = 2/3

two coordinates are (0,–2) and ()

Graph:

It can be seen from graph that there is breakage in curve at (0,0)

Thus, it is discontinuous at x = 0

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