Differentiabilty
The next important fundamental concept is differentiabilty or determining if a function is differentiable. In order for a function to be differentiable it must first be continuous (hence, must also have a limit) - if it is not continuous, it can't be differentiable. Secondly, the SLOPE of the function must be the same from both sides at a particular point.
More informally, a simple way to conceptually understand the notion of differentiability is only one tangent line can be drawn to the graph at any point or a graph must be smooth and can not have and sharp points, such as the absolute value function.
Please note that a function can be differentiable for all values on an interval, except for a single point.
Special note: To prove that a function is not differentiable, you must show that it is either not continuous or that lim x-> A+ f '(x) = lim x-> A- f '(x).
1. If a vertical asymptote exists at a point, it is NOT continuous and therefore not differentiable either.
2. The absolute value function is continuous at all points and differentiable at all points except at the bottom (or top if it is negative) of the v-shaped graph. Why? The slopes are different as you approach the "sharp v" from both sides.
More informally, a simple way to conceptually understand the notion of differentiability is only one tangent line can be drawn to the graph at any point or a graph must be smooth and can not have and sharp points, such as the absolute value function.
Please note that a function can be differentiable for all values on an interval, except for a single point.
Special note: To prove that a function is not differentiable, you must show that it is either not continuous or that lim x-> A+ f '(x) = lim x-> A- f '(x).
1. If a vertical asymptote exists at a point, it is NOT continuous and therefore not differentiable either.
2. The absolute value function is continuous at all points and differentiable at all points except at the bottom (or top if it is negative) of the v-shaped graph. Why? The slopes are different as you approach the "sharp v" from both sides.
Determining Differentiabilty from a Graph
Determining Differentiabilty Algebraically for a Piecewise Function I
Determining Differentiabilty Algebraically for a Piecewise Function II
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