Discuss the continuity of the function at the point x = 0.

__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.

Given,

…… equation 2

we are asked to check its continuity at x=0

∴ we need to check LHL, RHL and value of function at x = 0, if all comes out to be equal we can say f(x) is continuous at x=0 else it is discontinuous.

Clearly,

f(0) = 1 [from eqn 2]

LHL =

RHL =

Thus, LHL = RHL ≠ f(0)

∴ f(x) is discontinuous at x = 0

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