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Find the points of discontinuity, if any, of the following functions :
A real function f is said to be continuous at x = c, where c is any point in the domain of f if :
where h is a very small ‘+ve’ no.
i.e. left hand limit as x → c (LHL) = right hand limit as x → c (RHL) = value of function at x = c.
This is very precise, using our fundamental idea of limit from class 11 we can summarise it as, A function is continuous at x = c if :
Here we have,
Function is defined for all real numbers so we need to comment about its continuity for all numbers in its domain ( domain = set of numbers for which f is defined )
Function is changing its nature (or expression) at x = 1, So we need to check its continuity at x = 1 first.
f(1) = 4 [using eqn 1]
∴ f(x) is discontinuous at x = 1.
Let c be any real number such that c ≠ 0
f(c) = c3 – c2 + 2c – 2 [using eqn 1]
∴ f(x) is continuous for all real x except x =1