Continuity

Alexander Lorien

New member
Is the statement, "If 𝑓 is a continuous function at a, then so is |𝑓|." true or false? If true, give a formal proof. If false, provide a counterexample.

Subhotosh Khan

Super Moderator
Staff member
Is the statement, "If 𝑓 is a continuous function at a, then so is |𝑓|." true or false? If true, give a formal proof. If false, provide a counterexample.
What is the formal definition of a continuous function?

Please show us what you have tried and exactly where you are stuck.

Please follow the rules of posting in this forum, as enunciated at:

pka

Elite Member
Is the statement, "If 𝑓 is a continuous function at a, then so is |𝑓|." true or false? If true, give a formal proof. If false, provide a counterexample.
Do you know that $$||a|-|b||\le|a-b|~?$$
$$|a|=|a-b+b|\le|a-b|+|b|$$ implies $$|a|-|b|\le|a-b|$$
Likewise $$|b|-|a|\le|b-a|=|a-b|$$ or $$-|a-b|\le||a|-|b||\le|a-b|$$
So $$\large\left||a|-|b|\right|\le|a-b|$$

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Jomo

Elite Member
Can you draw some continuous functions for f(x) and see if |f(x)| is continuous? Make sure that your f(x) takes on both positive and negative values.

Post back with your results including pictures of the functions.

Alexander Lorien

New member
Does this mean that the statement is false??

Alexander Lorien

New member
Do you know that $$||a|-|b||\le|a-b|~?$$
$$|a|=|a-b+b|\le|a-b|+|b|$$ implies $$|a|-|b|\le|a-b|$$
Likewise $$|b|-|a|\le|b-a|=|a-b|$$ or $$-|a-b|\le||a|-|b||\le|a-b|$$
So $$\large\left||a|-|b|\right|\le|a-b|$$
Does it mean that the statement is false? Sorry i am confused

pka

Elite Member
Does it mean that the statement is false? Sorry i am confused
You simply need to sit down with an instructor. You need help.
If $$f$$ is a contusions at $$x=a$$ then if $$c>0$$ then $$\exists d>0$$ so that $$|x-a|<c\to |f(x)-f(a)|<d$$
BUT $$||f(x)|-|f(a)||\le|f(x)-f(a)| <d$$ Does that mean that mean that $$|f|$$ is continuous at $$x=a~?$$

Jomo

Elite Member
Does this mean that the statement is false??
Draw a few functions and see for yourself. You need to see if it is true or not. A few simple drawing will do wonders.