Continuity and differentiability
Understand the Problem
The question seems to be asking about the concepts of continuity and differentiability in mathematics, which are fundamental topics in calculus. Specifically, it likely seeks to understand how a function can be continuous and differentiable, as well as the relationship between the two concepts.
Answer
A function can be continuous and differentiable at a point, but must be continuous to be differentiable; $f(x) = |x|$ is continuous but not differentiable at $x = 0$.
Answer for screen readers
A function can be continuous and differentiable at a point, but a function must be continuous to be differentiable. A common example is $f(x) = |x|$, which is continuous but not differentiable at $x = 0$.
Steps to Solve
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Define Continuity A function $f(x)$ is continuous at a point $c$ if the following three conditions hold:
- The function $f(c)$ is defined.
- The limit $\lim_{x \to c} f(x)$ exists.
- The limit equals the function value: $\lim_{x \to c} f(x) = f(c)$.
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Define Differentiability A function $f(x)$ is differentiable at a point $c$ if the derivative $f'(c)$ exists. This is expressed mathematically as:
$$ f'(c) = \lim_{h \to 0} \frac{f(c + h) - f(c)}{h} $$- Differentiability implies continuity, meaning if $f(x)$ is differentiable at $c$, then it is also continuous at $c$.
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Illustrate the Relationship To understand the relationship:
- If a function is differentiable at a point, it must be continuous there.
- However, a function can be continuous at a point but not differentiable, exemplified by a sharp turn (cusp) at that point.
- Example Function One common example is the function $f(x) = |x|$.
- It is continuous everywhere, but it is not differentiable at $c = 0$ because the left-hand limit and right-hand limit of the derivative do not match.
A function can be continuous and differentiable at a point, but a function must be continuous to be differentiable. A common example is $f(x) = |x|$, which is continuous but not differentiable at $x = 0$.
More Information
Continuity ensures that a function does not have any jumps or breaks, while differentiability indicates that you can find a tangent line at that point. Understanding these concepts is crucial for analyzing functions in calculus.
Tips
- Confusing continuity with differentiability: Remember that differentiability implies continuity, but continuity does not imply differentiability.
- Not checking all conditions for continuity, such as ensuring limits are equal to the function value.
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