EDB — 1DT

Exercises

1. [1DT] What can you put in place of "???" so that the function

   \[ g(x) = \begin{cases} ??? & \text{if}~ ~ ~ 0{\lt}x{\lt}1~ , \\ {} 1 & \text{if}~ ~ x≥ 1 ~ ,\\ {} 0 & \text{if}~ ~ x≤ 0~ . \end{cases} \]

   is \(C^∞\)?

   More generally, how can two \(C^∞\) functions be connected, so that the whole function is \(C^∞\)? Given \(f_ 0,f_ 1∈ C^∞\), show ¹ that there is a function \(f∈ C^∞\) that satisfies

   \begin{eqnarray*} f(x) = f_ 0(x)& \text{if}~ ~ ~ x≤ 0 \quad ,\\ {} f(x) = f_ 1(x)& \text{if}~ ~ ~ x≥ 1 \quad . \end{eqnarray*}
   Solution 1

   [1DV]↺↻