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If Find whether f(x) is continuous at x = 1.
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
where h is a very small positive no (can assume h = 0.00000000001 like this )
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, function is continuous else it is discontinuous.
Let’s solve :
To check whether function is continuous at x=3 we need to check whether LHL = RHL = f(c)
As continuity is to be checked at x = 1 therefore c=1 (in equation 1)
As, ………eqn 2
From eqn 2 :
f(1) = 2
Using equation 2 –
Clearly, LHL = RHL = f(1) = 2
∴ f(x) is continuous at x=1