To determine whether a function is continuous or differentiable, you can follow these steps:

1. Continuity

To check if a function $f(x)$ is continuous at a point $x = a$:

• Step 1: Ensure that $f(a)$ is defined. The function must have a finite value at $x = a$.
• Step 2: Calculate the limit of $f(x)$ as $x$ approaches $a$ from both directions, i.e., $\lim_{x \to a^-} f(x)$ and $\lim_{x \to a^+} f(x)$.
• Step 3: Check if the two one-sided limits are equal. If they are, then: $\lim_{x \to a} f(x) = f(a)$
• If all these conditions are satisfied, then $f(x)$ is continuous at $x = a$. If these conditions are true for all points within a domain, then $f(x)$ is continuous on that interval.

2. Differentiability

To determine if $f(x)$ is differentiable at a point $x = a$:

• Step 1: Check if the function is continuous at $x = a$ (as differentiability requires continuity).
• Step 2: Find the derivative of $f(x)$, often denoted by $f'(x)$ or $\frac{df}{dx}$, using the definition: $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$
• Step 3: Confirm that the limit exists and is finite. If it does, then $f(x)$ is differentiable at $x = a$.
• If $f(x)$ is differentiable at every point in an interval, then it is differentiable on that interval.

Notes:

• Continuity does not imply differentiability: A function can be continuous at a point without being differentiable there (e.g., $|x|$ at $x = 0$).
• Differentiability implies continuity: If a function is differentiable at a point, it must be continuous there.

Would you like to explore any specific function or a more detailed example?

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1. How can I determine if a piecewise function is continuous?
2. What’s an example of a function that is continuous but not differentiable?
3. How do limits play a role in determining continuity and differentiability?
4. Can you explain the role of the derivative in confirming differentiability?
5. How can we check continuity and differentiability on intervals instead of single points?

Tip: Always check for continuity first, as differentiability requires it.