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Functions are continuous if they don't have any breaking point. For example, consider the graph of \sin { x }. The function doesn't change its behaviour at any point. Hence we can call this function a continuous function. Now consider the graph of \tan { x }. You will see that the graph breaks at some certain points, such as \frac { \pi }{ 2 }, \frac { 3 \pi }{ 2 }, and many more. This function is known as a non-continuous function. The continuity also depends upon the range of the function. The continuity of the function is checked on all values that are in the interval. Interval can either be closed or open. If open, then it means you need to check from the point after the starting interval point till the point which is before the ending interval point. However, if you are provided with a close interval then you have to check your function against the values that are in the interval which includes starting and ending point of the interval. A function f(x) is continuous at the open interval [a, b] if:

  • The function is right-continuous, i.e., \lim_{ x \rightarrow { a }^{ + } } f(x)
  • The function is left continuous, i.e., \lim_{ x \rightarrow { b }^{ - } } f(x)

If f is continuous at a closed interval [a, b], then f is bounded on that interval.

Continuity Study

The best way to find whether the function is continuous or not is by putting up limits to a specific point. Imagine a point "a" that is between the interval. The limit will approach in two directions, from the positive side to that point and from the negative side to that point. After finding both limits, check whether they are equal to each other or not. The limit will start from the starting point of the interval (supposing close interval) to the point "a" for the positive limit. In the case of the negative limit approach, the limit will start from the ending point of the interval to the point "a". If they are equal, the function is continuous and if they are not equal, that means the function isn't continuous.

Example

Study the continuity of f(x) = \left\{\begin{matrix} { x }^{ 2 } \quad if \quad x < 2 \\ 4 \quad if \quad x \geq 2 \end{matrix}\right at the interval [0, 4].

 

f(x) is left-continuous at x = 0 , since f(x) = { x }^{ 2 } and polynomial functions are continuous at all of \mathbb{R}.

f(x) is right-continuous at x = 4 , since f(x) = 4 and polynomial functions are continuous at all of \mathbb{R}.

Study the continuity at x = 2, as this is the interval of the piecewise function. For f(x)to be continuous, it needs to be continuous at this point.

f(2)= 4

\lim_{ x \rightarrow { 2 }^{ - } } { x }^{ 2 } = 4

\lim_{ x \rightarrow { 2 }^{ + } } 4 = 4

Hence, \lim_{ x \rightarrow 2 } f(x) = 4

Thus, f (x) is continuous in the interval [0, 4].

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Hamza

Hi! I am Hamza and I am from Pakistan. My hobbies are reading, writing and playing chess. Currently, I am a student enrolled in the Chemical Engineering Bachelor program.