Here we are talking about the physical intuition behind continuous functions. The opposite of something continuous (using this everyday usage of the word continuity) is called discrete. How many liters of water you drank yesterday.The time it takes you to read this sentence.Īnd I hope you'll agree the following things are not continuous:.The change in velocity over time of an airplane taking off.The distance between a car and its destination (specifically, the change of distance).For example, I'm pretty sure you'll agree that the following things are continuous: We know intuitively when something is continuous. It doesn't grow by leaps, but continuously. For example, the growth of a plant is continuous. In calculus, something being continuous has the same meaning as in everyday use. In fact, calculus was born because there was a need to describe and study two things that we consider "continuous": change and motion. Yes, exponential functions are continuous as they do not have any breaks, holes, or vertical asymptotes.The concept of continuous functions appears everywhere. i.e., limₓ → ₐ f(x) = f(a) Are Exponential Functions Continuous? What is Continuous Function Formula?Ī function f(x) is continuous at x = a when its limit exists at x = a and is equal to the value of the function at x = a. For example, f(x) = |x| is continuous everywhere. Which Function is Always Continuous?įor a function to be always continuous, there should not be any breaks throughout its graph. Hence, the square root function is continuous over its domain. The graph of a square root function is a smooth curve without any breaks, holes, or asymptotes throughout its domain. For example, \(g(x)=\left\\right.\) Describe the Continuity of Square Root Function. What is Piecewise Continuous Function?Ī continuous function is said to be a piecewise continuous function if it is defined differently in different intervals. The polynomial functions, exponential functions, graphs of sin x and cos x are examples of a continuous function over the set of all real numbers. The graph of a continuous function should not have any breaks. Mathematically, f(x) is said to be continuous at x = a if and only if limₓ → ₐ f(x) = f(a). Here are some topics that you may be interested in while studying continuous functions.įAQs on Continuous Function What is the Definition of Continuous Function?Ī continuous function is a function whose graph is not broken anywhere. The functions are NOT continuous at holes.The functions are NOT continuous at vertical asymptotes.For a function to be differentiable, it has to be continuous.It means, for a function to have continuity at a point, it shouldn't be broken at that point.A function is continuous at x = a if and only if limₓ → ₐ f(x) = f(a).Here are some points to note related to the continuity of a function. The values of one or both of the limits limₓ → ₐ₋ f(x) and limₓ → ₐ₊ f(x) is ± ∞. It is called "infinite discontinuity". Limₓ → ₐ f(x) exists (i.e., limₓ → ₐ₋ f(x) = limₓ → ₐ₊ f(x)) but it is NOT equal to f(a). It is called "removable discontinuity". Limₓ → ₐ₋ f(x) and limₓ → ₐ₊ f(x) exist but they are NOT equal. It is called "jump discontinuity" (or) "non-removable discontinuity". Infinite discontinuity occurs at vertical asymptotes.Removable discontinuity occurs at holes.From the figures below, we can understand that By the definition of the continuity of a function, a function is NOT continuous in one of the following cases. We can see all the types of discontinuities in the figure below. There are different types of discontinuities as explained below. i.e., the graph of a discontinuous function breaks or jumps somewhere. i.e., over that interval, the graph of the function shouldn't break or jump.Ī function that is NOT continuous is said to be a discontinuous function. You can understand this from the following figure.Ī function is said to be continuous over an interval if it is continuous at each and every point on the interval. 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. 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.
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