Question

A function f(x) is said to be continuous at a point x = a when
a) f(x) is defined at a

b) f(x) tends to a unique finite limit as x → a-

c) f(x) tends to a unique finite limit as x a+
d) all of these​

Answer 1

Answer:

A function f(x) is said to be continuous at a point x = a, in its domain if the following three conditions are satisfied:

f(a) exists (i.e. the value of f(a) is finite)

Limx→a f(x) exists (i.e. the right-hand limit = left-hand limit, and both are finite)

Limx→a f(x) = f(a)

The function f(x) is said to be continuous in the interval I = [x1,x2] if the three conditions mentioned above are satisfied for every point in the interval I.

However, note that at the end-points of the interval I, we need not consider both the right-hand and the left-hand limits for the calculation of Limx→a f(x). For a = x1, only the right-hand limit need be considered, and for a = x2, only the left-hand limit needs to be considered.

Answer 2

⭕In other words, a function f is continuous at a point x=a, when

(i) the function f is defined at a

(ii) the limit of f as x approaches a from the right-hand and left-hand limits exist and are equal

(iii) the limit of f as x approaches a is equal to f(a).

⭕Answer for your question is all of the above sentences.