Question

1. the graph of a function f is shown above. at which value of x is f continuous, but not differentiable?

Explanation:

Step1: Recall the definition of continuity

A function \(y = f(x)\) is continuous at a point \(x = c\) if \(\lim_{x
ightarrow c^{-}}f(x)=\lim_{x
ightarrow c^{+}}f(x)=f(c)\). Visually, the graph has no breaks or holes at a continuous point.

Step2: Recall the definition of differentiability

A function \(y = f(x)\) is differentiable at a point \(x = c\) if the left - hand derivative \(f^{\prime}_{-}(c)=\lim_{h
ightarrow0^{-}}\frac{f(c + h)-f(c)}{h}\) and the right - hand derivative \(f^{\prime}_{+}(c)=\lim_{h
ightarrow0^{+}}\frac{f(c + h)-f(c)}{h}\) are equal. Geometrically, the function has a smooth curve (no sharp corners) at a differentiable point.

Step3: Analyze each point

• Point \(a\): The function has a break (discontinuity) at \(x = a\). So it is neither continuous nor differentiable at \(x=a\).
• Point \(b\): The function has a hole (discontinuity) at \(x = b\). So it is neither continuous nor differentiable at \(x = b\).
• Point \(c\): The function is continuous at \(x = c\) (the graph has no break). But it has a sharp corner at \(x = c\). For a differentiable function, the graph should be smooth. The left - hand and right - hand derivatives at \(x = c\) are not equal.
• Point \(d\): The function has a break (discontinuity) at \(x = d\). So it is neither continuous nor differentiable at \(x = d\).
• Point \(e\): The function has a break (discontinuity) at \(x = e\). So it is neither continuous nor differentiable at \(x = e\).

Answer:

C. \(c\)