Study the continuity of this function

However, the product of a continuous function and a discontinuous function is continuous. So, the given function is continuous on all real numbers.
  • #1
Felafel
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0

Homework Statement



f(x) = [x^2]sinπx, x ∈ R, being [x] the integer part of x


Homework Equations





The Attempt at a Solution



I'd say it's trivially continuos on all R, because it's the product of two continuos functions: y: = [x^2] and g: = sinπx.
However, being part of my analysis class homework (always a bit tricky to solve), it seems a bit too easy to me. So, i thought there might be something strange I haven't noticed. Or is my reasoning just correct?
thanks in advance!
 
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  • #2
Felafel said:

Homework Statement



f(x) = [x^2]sinπx, x ∈ R, being [x] the integer part of x


Homework Equations





The Attempt at a Solution



I'd say it's trivially continuos on all R, because it's the product of two continuos functions: y: = [x^2] and g: = sinπx.
However, being part of my analysis class homework (always a bit tricky to solve), it seems a bit too easy to me. So, i thought there might be something strange I haven't noticed. Or is my reasoning just correct?
thanks in advance!

x^2 is continuous. The integer part of x^2 is not continuous.
 

FAQ: Study the continuity of this function

What does it mean to study the continuity of a function?

The continuity of a function refers to its behavior and smoothness across its entire domain. It is a measure of how well the function can be traced without any abrupt changes or gaps in its graph.

Why is studying the continuity of a function important?

Studying the continuity of a function is important because it helps us understand the behavior of the function and its graph. It also allows us to determine if the function is well-defined and has predictable behavior.

How do you determine if a function is continuous?

A function is continuous if it is defined at every point in its domain and there are no abrupt changes or gaps in its graph. This means that the limit of the function exists at every point and is equal to the value of the function at that point.

What are the three types of continuity?

The three types of continuity are:
1. Pointwise continuity - when the limit of the function exists at each point in its domain
2. Uniform continuity - when the function remains continuous over the entire domain, without any abrupt changes or gaps
3. Differentiability - when the function has a well-defined derivative at each point in its domain

How can discontinuities in a function be classified?

Discontinuities in a function can be classified into three types:
1. Removable discontinuities - when the function has a hole or a gap at a specific point, but can be made continuous by redefining the function at that point
2. Jump discontinuities - when the function has a sudden jump or change at a specific point
3. Infinite discontinuities - when the function approaches infinity or negative infinity at a specific point

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