These two conditions together will make the function to be continuous (without a break) at that point. "limₓ → ₐ f(x) exists" means, the function should approach the same value both from the left side and right side of the value x = a and "limₓ → ₐ f(x) = f(a)" means the limit of the function at x = a is same as f(a). Is this definition really giving the meaning that the function shouldn't have a break at x = a? Let's see. A function f(x) is continuous at a point x = a if The mathematical definition of the continuity of a function is as follows. Question 5: Is a point function continuous?Īnswer: A point function is not continuous according to the definition of continuous function.A function f(x) is said to be a continuous function in calculus at a point x = a if the curve of the function does NOT break at the point x = a. In a removable discontinuity, one can redefine the point so as to make the function continuous by matching the particular point’s value with the rest of the function. Question 4: What is meant by discontinuous function?Īnswer: Discontinuous functions are those that are not a continuous curve. Symbolically, one can write this as f (x) = 6. For example, given the function f (x) = 3x, the limit of f (x) as the approaching of x takes place to 2 is 6. Question 3: What is limit with regards to continuity?Īnswer: A limit refers to a number that a function approaches as the approaching of an independent variable of the function takes place to a given value. The limit of the function as the approaching of x takes place, a is equal to the function value f(a).The limit of the function as the approaching of x takes place, a exists.Question 2: Explain the three conditions of continuity?Īnswer: The three conditions of continuity are as follows: Continuous functions are very important as they are necessarily differentiable at every point on which they are continuous, and hence very simple to work upon. This concludes our discussion on the topic of continuity of functions. Thus all the three conditions are satisfied and the function f(x) is found out to be continuous at x = 1. In this type of discontinuity, the right-hand limit and the left-hand limit for the function at x = a exists but the two are not equal to each other. On the basis of the failure of which specific condition leads to discontinuity, we can define different types of discontinuities. If any one of the three conditions for a function to be continuous fails then the function is said to be discontinuous at that point.
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