Answer to Question #5258 in Calculus for Anil

Question

Give an example of non differentiable function which is continuous everywhere.

Answer 1

1). One example is f(x) = |x|. It's continuous everywhere on R, but not differentiable at x = 0. The derivative to the left is -1 and to the right is 1 So, f is not differentiable at x = 0.
<img src="../../..//assignments/uploaded_files/static/92/4f/924f199a3d32805a65cb4b25a4ce9079.png" title="" alt="">
2). Another very famous example is Weierstrass functions. They are continuous everywhere but differentiable nowhere.
For example:
<img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAATMAAABDCAIAAAB/b3V8AAAEn0lEQVR4nO2d24GjMAxFqYuCXA/V0AzFeD8SEr9kMIiM1j7naxIexhddWwbNzOQBwB7TX18AABTAmQAWwZkAFsGZABbBmQAWwZkAFsGZABbBmQAWwZkAFsGZABbBmQAWwZkAFsGZABbBmQAWwZkAFsGZABbBmQAWwZkAFsGZABbBmVqsbpqX7a+vwhZoch2ceZnVTW713nu/LfM0TdP+0e9fDhaVX0F8SZPxBLkFzrzGNwrf88LqEmf6JFQ7J+qrpMlIgtwFZ16hEGElZ44TiuV+FjQZRZD74Mx2imlZ2Zl+W+b+Q1HKU0uaDCGIBjizmdPzw2dD58srcR4U84jOBVEBZzYijfmiMytbuqAyCQo971wQJXBmG2IcVsKt60isZadSx7sWRAuc2cS2zEIqtrpJTNI6jkRZEF/RpGNB1OjEmdUA0QuEYjOrmyLy66hfnf4VHfKsIP5Qk34F0aMDZ567FyraXz3JMzf+XnwjSIIxd/73zmyoLLn9UHBb5ms374k5Iu94MFGda6xzQfYypKw8S8TSY2Orzjw7gDVpefgu7eBstbVknfPj8cWOR6F++jo7F+Rz3M8EUcSqM0/SKNTR7o8GouodrV7p+RmpW0ESCU4rclMQRdqc+cqXwpHotr5hyvE61/ubvZXXx3nZPulaVJ+ZDnHB7u8Dgh2ObtCJQLy2EikEonbHpascU5C0Y8H2JwVRpMGZ2zK7NZa0UP0R6Xv00DISYnXheV8/v9YO2zJPzr3aim5BptO2uFcl9euoLGqOI+1Hgajd8f2cqdLDChI38M1tnxVEkfZsNhp93N2rLN7bdC0fDmyRNIJO0aiYNpd+3TCU5BcrHVnqZh456h0PjsrC/jeC+JOa/ECQ9M2NmEjoCbItczxvJRN+yyjW7szv4LY6lYfMcYrs82V4+DkWThqzpYW8peTtkY5LzSFI/tuzDwjyt87cOyVMmG3ZbHiUlCOH21JBK1mdMCBW1fn9Aw/VjkfNnZ0hBhAkc8Vjgihy5dns6qbZuduJbLieiFcZ0+TW78ZAjH2372ydK7W693ok3nE/3MRLAv2Ohw0kgTimIB/ieezT2DOCKHLFmVqviaOFQLJaj58Ahwv9POEppQ/JSfbNB5d9sIfWi3X1jgdHT+kphhVk3ycdwJ4URJFr2ayhIqYT41zDXgcIuU4QWOXbdn09VuNel54U5LD+pltB9Gh2ZvBE2xD1tOp60pVQyBbOFN88UYvmqy0+clSBcs8O62/6FUSP884szf7DcRBSwuZnZggTlHp8XH/TsSBq/OfVeT+nlvJI8dZ1HBYEqdTfCF9ADs5spGTNUvHNl87jsCCIVH8TbO5YECVwZjOr+IcZ8+Ib/8uneX9FQZBa/U3/gqiAM9vJH89/KBaddT8/VATxhVeX/QuiAc68ghhdaaI2ShjK/czKHsYQ5D448xqfCKsU3wwVhsW+pvU3IwlyF5x5mT3MhJfq9RSvR9bkX5ski+7xBLkFzgSwCM4EsAjOBLAIzgSwCM4EsAjOBLAIzgSwCM4EsAjOBLAIzgSwCM4EsAjOBLAIzgSwCM4EsAjOBLAIzgSwyD/zyh05dxRaLQAAAABJRU5ErkJggg==" alt="">
<img style="width: 264px; height: 126px;" src="data:image/png;base64,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" alt="">

It’s not smooth as differentiable function.