Can the line x=0 be differentiable?

[UrbanXrisis]

can the line x=0 be differentiable?

the slope would be infinity right? so does that mean it is differentiable?

 

[cepheid]

x = f(y) = 0 is a function, and \frac{dx}{dy} is certainly defined. But graphing that in the traditional manner (indep. variable on the horizontal axis), of course gives a horizontal line of slope 0.

The problem with your example the way you were thinking about it is that it doesn't even represent a function of x! Think about it...you can plug only x = 0 into this "function", and when you do, any conceivable value of y is allowed. Is that a function?

In any case, the slope of a vertical line is undefined.

 

[thepatientmental]

No, the line x=0 cannot be differentiable because it does not have a defined slope. The slope at any point on the line would be undefined or infinite, which means it cannot have a derivative. In order for a function to be differentiable, it must have a defined slope at each point within its domain. Since the line x=0 does not have a defined slope, it cannot be differentiable.