Continuous Function Homework: Determine & Sketch

In summary, the function f(t) = {10-t, 0<=t<=8 and 10, 8<=t<=10) is piecewise continuous on the segment [0, 10] as it is continuous on the interior of each of its pieces. It is not equal to neither as it is not continuous on the entire segment.
  • #1
kieranl
24
0

Homework Statement



Determine whether the function is continuous, piecewise continuous or neither on the segment [0, 10] and sketch the graph of f(t).

f(t)= {10-t, 0<=t<=8 and 10, 8<=t<=10)

The Attempt at a Solution



I would say that it was neither as the right hand limit at t = 8 doesn't equal the left hand limit. But I am not sure wat the difference is between a continuous function and a piecewise continuous function?

thanks for any help
 
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  • #2
You are absolutely right. A piecewise continuous function is continuous on the interior of each of the pieces, a continuous function is continuous everywhere.
 
  • #3
No, Dick, NOT "absolutely right". That function is equal to 10- t for [itex]0\le t\le 8[/itex] and 10 for [itex]8\le t\le 10[/itex] so it is continuous on the interior of each of those intervals and is "piecewise continuous", not "neither".
 
  • #4
HallsofIvy said:
No, Dick, NOT "absolutely right". That function is equal to 10- t for [itex]0\le t\le 8[/itex] and 10 for [itex]8\le t\le 10[/itex] so it is continuous on the interior of each of those intervals and is "piecewise continuous", not "neither".


Right. I only saw that 'the limit doesn't exist' and missed the 'neither'. Thanks.
 

FAQ: Continuous Function Homework: Determine & Sketch

What is a continuous function?

A continuous function is a type of mathematical function where there are no sudden or abrupt changes in the output value as the input value changes. This means that the graph of a continuous function has no breaks, holes, or jumps.

How do I determine if a function is continuous?

To determine if a function is continuous, you can use the following criteria:

  • The function is defined at the point in question.
  • The limit of the function as x approaches the point exists.
  • The limit of the function as x approaches the point is equal to the function value at the point.

What does it mean to sketch a continuous function?

To sketch a continuous function means to create a graph of the function that accurately represents the behavior of the function for all values of the input variable. This includes showing any breaks, holes, or jumps in the graph.

How do I sketch a continuous function?

To sketch a continuous function, you can follow these steps:

  • Determine the domain and range of the function.
  • Identify any breaks, holes, or jumps in the graph.
  • Plot points to create a rough sketch of the graph.
  • Use the behavior of the function to refine and complete the graph.

Why is it important to understand continuous functions?

Continuous functions are essential in many areas of mathematics and science. They are used to model real-world phenomena and make predictions about the behavior of systems. Understanding continuous functions is also important in calculus and other advanced mathematical concepts.

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