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Let $f(x)=\begin{cases} x+c& \text{if } x \leq c,\\ 1 &\text{if x > c} \end{cases}$

(a) Find a value for c so that the function f is continuous.

(b) Is the function f differentiable at c? Explain.

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    Welcome to MSE. What is your question about the excercise you wrote above? What have you tryed and where did you get stuck? – 2017-02-18
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    Maybe you should begin by drawing graphs for different values of $c$ to have an idea of your problem. – 2017-02-18

1 Answers 1

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For $f$ to be continuous we want $f(c) = 1$. So we want $2c = 1$ so $c=1/2$.

For $f$ to be differentiable at $c$ we need the left and right derivatives to be the same. Coming from the left $f' = 0$ and coming from the right $f' = 1$ therefore it's not possible for $f$ to be differentiable at $c$.