Question

A function /is said to be continuous for x ∈ R, if

(A) unique and distinct
(B) continous for x ∈ R
(C) not differentiable for x ∈ R
(D) differentiable for x ∈ R

Answer 1

Answer:

A function f : A → R is continuous if it is continuous at every point of A, and it is continuous on B ⊂ A if it is continuous at every point in B. The definition of continuity at a point may be stated in terms of neighborhoods as follows. x ∈ A ∩ U implies that f(x) ∈ V .

Answer 2

Answer:

A function /is said to be continuous for x ∈ R, if differentiable for  x ∈ R

Step-by-step explanation:

Definition of Continuity

• When x=c, function f is continuous, which is equivalent to declaring that the function's two-side limit exists and equals to f (c).
• A function f(x) is said to be continuous at a point x = a, in its domain.

Continuous function should satisfy the below three requirements:

• The equation for the function is x = a.

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

• The function has a limit as x gets closer; this limit is indicated by the                presence of a.

           $\lim_{x \to \ a} f(x)$ exists (both values are finite)

• As x approaches, the limit of the function, a, equals the value of the function, f. (a).

          $\lim_{x \to \ a} f(x) = f(a)$

Properties of Continuity:

• The set of all real numbers contains all continuous polynomial functions.
• The set of all real numbers is continuous under the absolute value function |x|.
• At all real numbers, exponential functions are continuous.

Final answer:

If a function f(x) is differentiable ∀ x ∈ R, then it will be continuous function ∀ x ∈ R

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