Question

If $f(x)$ is differentiable function in x then it is

• A. \( \text{Unbounded} \)
• B. \( \text{Many Valued} \)
• C. \( \text{Continuous} \)
• D. \( \text{Bounded} \)

Hint:

Differentiability implies continuity. Think of a smooth curve (differentiable) versus a curve with breaks or sharp turns (continuous but not differentiable).

Answer 1

The Correct Option is C

A fundamental theorem in calculus states that if a function $f(x)$ is differentiable at a point $x$, then it must also be continuous at that point. Differentiability is a stronger condition than continuity. If a function has a derivative at a point, it means that the function is "smooth" at that point, without any breaks, jumps, or sharp corners. The converse is not necessarily true; a function can be continuous at a point but not differentiable (e.g., $f(x) = |x|$ at $x = 0$). Therefore, if $f(x)$ is a differentiable function in $x$, then it must be continuous in $x$. It does not necessarily have to be unbounded, many-valued (a function by definition has a single output for each input), or bounded.