Final answer:
A function must be continuous at a point to be differentiable at that point, so a function that is differentiable at x=c is necessarily continuous at x=c. This fact is fundamental in calculus and the study of functions, making the statement true.
Step-by-step explanation:
When considering the differentiability of a function at a point x=c, it is important to understand that differentiability implies continuity. A function f(x) that is differentiable at x=c must have a defined, single derivative at that point. For a derivative to exist, the function must not have any abrupt changes, breaks, or holes at x=c. In other words, the function must be smooth at that point, which means it is also continuous there. By definition, if a function has a derivative at a certain point, then it must be continuous at that point as well.
Examining the given statement, a function that is differentiable at x=c must be continuous at x=c, we can confidently state that this is a true statement. Differentiability is a stronger condition than continuity, which means that all differentiable functions are continuous, but not all continuous functions are necessarily differentiable. So, if a function has a first derivative that exists at x=c, then the function must also be continuous at that point.
For a graphical understanding, imagine the plot of a differentiable function around x=c. The graph would show a smooth curve without any jumps or gaps, confirming continuity. Any function that exhibits a discontinuity at a point cannot have a derivative at that point since the slope of the tangent, which the derivative represents, would not be defined.
Therefore, to the student's original question:
A function that is differentiable at x=c must be continuous at x=c: a) True