Piece-Wise Function Continuity

If they are not, then the function is discontinuous at that point. In summary, To determine if a function is continuous, you need to check if the function exists at the given point, if the limit as x approaches that point exists, and if the limit equals the function value at that point. If any of these conditions fail to hold, the function is discontinuous at that point. In this case, you will need to calculate the limit at x=-2 and see if the left and right sided limits are equal. If they are not, then the function is discontinuous at x=-2.
  • #1
justapinkday
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Homework Statement


Determine whether the given function is continuous. If it is not, identify where it is discontinuous and which condition fails to hold.

f(x)= {x^3 if x < or = -2
{2 if x > -2


Homework Equations


The conditions are that a function is said to be continuous at x=c if the following conditions hold:
1 f(c) exists
2 lim as x approaches c f(x) exists
3 lim as x approaches c f(x) = f(c)



The Attempt at a Solution


My professor has not explained this and has assigned it...no clue what to do!
 
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  • #2
You will have to calculate the limit in each point and see whether the left and right sided limits are equal.
 

Related to Piece-Wise Function Continuity

What is a piece-wise function?

A piece-wise function is a mathematical function that is defined by different formulas or rules on different intervals or "pieces" of its domain. This allows for the function to be continuous, or smooth, over its entire domain, even if the individual pieces are not continuous.

How is continuity defined for a piece-wise function?

A piece-wise function is considered continuous if it is continuous at each point within each piece and if the limit of the function as it approaches the boundary between pieces is equal to the value of the function at that boundary point.

What is the significance of piece-wise function continuity?

Piece-wise function continuity allows for a more flexible and accurate representation of real-world phenomena in mathematical models. It also allows for the use of different equations or rules on different intervals, which can make it easier to solve complex problems.

What are some common examples of piece-wise functions?

Some common examples of piece-wise functions include the absolute value function, the floor and ceiling functions, and the step function. These functions are often used to model real-world situations such as distance traveled, temperature changes, or population growth.

How can I determine if a piece-wise function is continuous?

To determine if a piece-wise function is continuous, you can evaluate the function at each boundary point between pieces. If the limit of the function as it approaches that point is equal to the value of the function at that point, then the function is continuous. Additionally, you can graph the function and look for any discontinuities or "jumps" in the graph.

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