Discuss the continuity of the following functions at the indicated point(s).

at x = 0

Idea to solve problem :

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 or even smaller )

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 the problem now:

we need to check continuity at x = 0 for given function

……equation 2

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

Clearly,

f(0) = 0 [from equation 2]

LHL = [∵ cos (–θ) = cos θ]

RHL = [from eqn 2]

Why

Since whatever is value of h, cos(1/h) is goimg to range from –1 to 1

As h→ 0 i.e. h is approximately 0

∴ 0*some finite quantity is equal to 0.

Clearly, LHL = RHL = f(0)

∴ f(x) is continuous at x = 0

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