true or false? are all differentiable functions continuous 

True. All differentiable functions are continuous. 

If a function \( f(x) \) is differentiable at a point \( a \), then it must also be continuous at that point. Differentiability implies continuity because the definition of the derivative involves a limit that requires the function to approach a single value as the input approaches the point of differentiation.

Would you like more details or have any questions?

Here are 5 questions you might consider next:
1. What is the formal definition of a differentiable function?
2. Can you explain why differentiability implies continuity?
3. Are all continuous functions differentiable?
4. What are some examples of functions that are continuous but not differentiable?
5. How do you prove a function is differentiable at a point?

Tip: Remember that while differentiability implies continuity, the converse is not necessarily true. There are continuous functions that are not differentiable at certain points.

Explore the relationship between differentiable and continuous functions. Learn why all differentiable functions are continuous and delve into the mathematical definition and implications. Suitable for advanced high school and college students.

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