A function is differentiable at a point if it satisfies certain conditions. Differentiability implies the existence of a derivative at that point, and it also means the function is smooth (without sharp corners or cusps) at that point. Here are the key conditions for differentiability at a point $x = c$:

1. Continuity at the point $x = c$:

• For a function $f(x)$ to be differentiable at $x = c$, it must be continuous at that point. In other words: $\lim_{x \to c} f(x) = f(c)$
• If the function has a discontinuity at $x = c$, it cannot be differentiable there.

2. Existence of the derivative:

• The derivative must exist at $x = c$. This means the limit that defines the derivative must exist: $f'(c) = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h}$
• If this limit exists and is finite, the function is differentiable at $x = c$.

3. No sharp corners or cusps:

• If the function has a sharp corner or a cusp at $x = c$, the derivative does not exist at that point, and the function is not differentiable.
• For example, the absolute value function $f(x) = |x|$ has a sharp corner at $x = 0$, so it is not differentiable at $x = 0$, even though it is continuous there.

4. No vertical tangent:

• The derivative must not approach infinity at $x = c$. A vertical tangent (where the slope becomes infinite) implies that the derivative does not exist at that point.

5. Both left-hand and right-hand derivatives must be equal:

• The left-hand derivative (approaching from the left) and the right-hand derivative (approaching from the right) at $x = c$ must both exist and be equal: $\lim_{h \to 0^-} \frac{f(c+h) - f(c)}{h} = \lim_{h \to 0^+} \frac{f(c+h) - f(c)}{h}$
• If these two limits are not equal, the function is not differentiable at $x = c$.

In summary, a function is differentiable at $x = c$ if it is continuous there, smooth (no sharp corners or cusps), and the left-hand and right-hand derivatives exist and are equal.

---

Do you have any questions, or would you like further details on a specific condition?

Here are 5 related questions to explore:

1. What is the difference between continuity and differentiability?
2. Can a function be continuous but not differentiable? Provide examples.
3. How can we determine if a piecewise function is differentiable at a boundary point?
4. What is the geometric interpretation of differentiability?
5. How does the concept of differentiability extend to higher dimensions?

Tip: Differentiability implies continuity, but continuity does not necessarily imply differentiability. Always check for smoothness when verifying differentiability!