Why are these false?
If a function has derivatives from both the right and left at a point then its differentiable at that point.
are they the same?
These statements are false because having derivatives from both the right and left at a point does not necessarily guarantee that the function is differentiable at that point.
To understand why, let's first discuss the definitions of derivatives and differentiability. The derivative of a function at a particular point represents the slope of the tangent line to the graph of the function at that point. A function is said to be differentiable at a point if and only if its derivative exists at that point.
Now, if a function has derivatives from both the right and left at a point, it means that the one-sided derivatives exist. The right-handed derivative is the derivative calculated from the right side of the point, while the left-handed derivative is calculated from the left side.
However, differentiability requires not only the existence of one-sided derivatives but also that they are equal. In other words, the right-handed derivative and the left-handed derivative should be the same at that point. This condition ensures that there is a unique tangent line, and the graph does not have any discontinuities or sharp turns at that point.
So, even if a function has derivatives from both the right and left at a point, if the one-sided derivatives are not equal, the function is not differentiable at that point. In such cases, the function may have a sharp corner or a jump discontinuity, rendering it non-differentiable.
To determine if the function is differentiable at a point, you need to calculate both the right-handed and left-handed derivatives and compare them. If they are equal, then the function is differentiable. Otherwise, it is not differentiable at that point.