Consider {{MATH_56}} What values of {{MATH_57}} and {{MATH_58}} make {{MATH_59}} differentiable at {{MATH_60}}? {{CMD_61}} To start, we know that {{MATH_62}} must be continuous at {{MATH_63}}, since it has to be differentiable there. We will start by making {{MATH_64}} continuous at {{MATH_65}}. Write with me: {{ENV_66}} So for the function to be continuous, we must have {{MATH_67}} We also must ensure that the function is differentiable at {{MATH_68}}. In other words, we have to ensure that the following limit exists {{MATH_69}} In order to compute this limit, we have to compute the two one-sided limits {{MATH_70}} and {{MATH_71}} since {{MATH_72}} changes expression at {{MATH_73}}. Write with me {{ENV_74}} and {{ENV_75}} Hence, we must have {{MATH_76}} Ah! So now the equation that must be satisfied {{ENV_77}} Therefore, {{MATH_78}}. Thus setting {{MATH_79}} and {{MATH_80}} will give us a function that is differentiable (and hence continuous) at {{MATH_81}}. {{CMD_82}} {{CMD_83}} Can we tell from its graph whether the function is differentiable or not at a point {{MATH_84}}?