If f is differentiable at x=c, which of the following statements must be true?
A. f is continuous at x=c
B. f^1 is continuous at x=c^1
C. f is continuous at y=c
D. f is continuous at y^1=c
To determine which statements must be true if a function f is differentiable at x = c, we can refer to the definition of differentiability.
A function f is said to be differentiable at x = c if the following limit exists:
lim┬(h→0)〖(f(c+h)-f(c))/h 〗
Now let's examine each statement and see if it must be true based on the differentiability of f at x = c:
A. f is continuous at x=c:
This statement must be true. Differentiability implies continuity at x = c. In other words, if a function is differentiable at a point, it has to be continuous at that point.
B. f^1 is continuous at x=c^1:
Here, f^1 denotes the derivative of f. This statement may or may not be true. The differentiability of f at x = c does not guarantee that its derivative f^1 is continuous at x = c. Although differentiability implies the existence of the derivative, it does not guarantee its continuity.
C. f is continuous at y=c:
This statement is not relevant to the differentiability of f at x = c. The variable y does not play a role in determining differentiability or continuity.
D. f is continuous at y^1=c:
Similar to statement C, this statement is also not relevant to the differentiability of f at x = c. The variable y^1 does not play a role in determining differentiability or continuity.
In conclusion, for a function f that is differentiable at x = c, the statement that must be true is A: f is continuous at x = c.