A function is continuous at a point if the limit exists, the function is defined, and the limit equals the function value at that point.

A function is differentiable at a point if its derivative exists at that point, meaning the left-hand and right-hand derivatives are equal.

A cusp is a point on a curve where the tangent line changes direction abruptly, and the derivative is undefined.

A corner is a point on a graph where the left and right derivatives are different, resulting in a sharp turn.

A vertical tangent is a tangent line to a curve that is vertical, indicating that the derivative approaches infinity at that point.

A jump discontinuity occurs when the left and right limits of a function at a point exist but are not equal.

A removable discontinuity is a point on a graph that is not continuous, but can be made continuous by redefining the function at that point.

Differentiability implies continuity. If a function is differentiable at a point, it must also be continuous at that point.

The limit of the difference quotient as x approaches a from the left.

The limit of the difference quotient as x approaches a from the right.

A sharp corner indicates that $f'(x)$ is undefined at that point, as the left-hand and right-hand derivatives are not equal.

A vertical tangent indicates that $f'(x)$ approaches infinity or negative infinity, meaning the derivative is undefined at that point.

Look for discontinuities (jumps, holes), sharp corners, cusps, or vertical tangents. These are all points where the function is not differentiable.

The derivative does not exist at the point of jump discontinuity.

The derivative does not exist at the point of removable discontinuity.

If a tangent line exists and has a finite slope, the function is differentiable at that point. If the tangent line is vertical or doesn't exist, the function is not differentiable.

A cusp indicates that the derivative is undefined at that point.

A corner indicates that the derivative is undefined at that point.

A vertical tangent indicates that the derivative is undefined at that point.

A discontinuity indicates that the derivative is undefined at that point.

1. Check for continuity at the transition point. 2. Find the derivatives of each piece. 3. Evaluate the left-hand and right-hand derivatives at the transition point. 4. Compare the values; they must be equal for differentiability.

1. Recognize that absolute value functions often have a corner. 2. Check the left and right-hand derivatives. 3. Since the derivatives are different, $f(x)$ is not differentiable at $x=a$.

1. Look for discontinuities (jumps, holes, vertical asymptotes). 2. Look for sharp corners or cusps. 3. Look for vertical tangents.

1. Find the left-hand limit of the derivative. 2. Find the right-hand limit of the derivative. 3. Check if both limits exist and are equal. 4. Check if f'(a) exists and equals the limits.

1. Determine if the discontinuity is removable. 2. If removable, redefine the function to be continuous. 3. Check the left and right-hand derivatives at the point. 4. If the derivatives match, it is differentiable.

1. Check for any discontinuities. 2. Check for any sharp corners or cusps. 3. Check for any vertical tangents. 4. If none of the above exist, the function is differentiable.

1. Check if function is continuous. 2. Find the derivative of the function. 3. Check if the derivative exists at the given point. 4. If the derivative exists, the function is differentiable.

1. Check if function is continuous at transition point. 2. Find the derivative of each piece. 3. Check if the left-hand and right-hand derivatives are equal at the transition point. 4. If the derivatives are equal, the function is differentiable.

1. Rewrite the function as a piecewise function. 2. Find the derivative of each piece. 3. Check if the left-hand and right-hand derivatives are equal at the point where the absolute value is zero. 4. If the derivatives are equal, the function is differentiable.

1. Find the derivative of the function. 2. Check if the derivative approaches infinity at the given point. 3. If the derivative approaches infinity, the function is not differentiable.