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armolinasf
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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.
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.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.
Continuity of a function means that there are no sudden jumps or breaks in the graph of the function. In other words, the function is defined and has a value at every point on its domain without any gaps or holes.
The three conditions for a function to be continuous are: the function is defined at the point in question, the limit of the function at that point exists, and the limit and the value of the function at that point are equal.
To show continuity at a specific point, you need to evaluate the function at that point and then take the limit of the function as it approaches that point. If the limit and the value of the function at that point are equal, then the function is continuous at that point.
Yes, a function can be continuous at one point and not at another. A function can be continuous at a point if it satisfies the three conditions for continuity, but may not be continuous at other points if it fails to meet any of the conditions.
Some common examples of continuous functions are polynomial functions, exponential functions, logarithmic functions, sine and cosine functions, and rational functions (as long as the denominator is not equal to zero).