Showing that a function is continuous

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To demonstrate that a function is continuous, one must show that the limit of the function as it approaches a point equals the function's value at that point. Continuity does not require the function to be differentiable, as some functions, like y = |x|, can be continuous at points where they are not differentiable. The definition of continuity involves the graph being an unbroken curve over an interval. It is essential to check the limit condition, specifically that lim_{x to a} f(x) = f(a). Understanding these concepts is crucial for proving continuity effectively.
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Homework Statement


How would I demonstrate that a function is continuous? Would I just show that it's derivative exists? Thanks for the help.
 
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armolinasf said:

Homework Statement


How would I demonstrate that a function is continuous? Would I just show that it's derivative exists? Thanks for the help.
But what if the function is not differentiable? There are functions that are not differentiable at a given point that happen to be continuous there. For example, y = |x| at x = 0.
 
Then it would be where it's undefined? Also, would I have to examine its end behavior?
 
No. My example function, f(x) = |x|, is defined everywhere.

How is continuity defined in your book?
 
It's defined by saying that a function is continuous if its graph is an unbroken curve over an interval
 
The way continuity at a point is usually defined is as a limit:
A function f is continuous at a number a in its domain iff
\lim_{x \to a}f(x) = f(a)
 

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